Clbc lilnivcrsit^ of CbicaQO 
ICibrarics 



F-e4 




GIFT OF 



THE UNIVERSITY OF CHICAGO 



INDEXINa A iffil\lTAL CHAMCTERISTIC 



A DISSERTATION 
SUBMITTSI' TO THE FACULTY 
OF THE GRADUATE SCHOOL OF ARTS AMD LITEPJITURE 
IN CANDIDACY FOR THE DEGREE OF 
DOCTOR OF PHILOSOPHY 



DEPARTMENT OF EDUCATION 



BY 
KARL JOHN HOLZINGER 

n 

CHICAGO, ILLINOIS 
SEPTEMBER, 1922 



"ffW 









1 



r^ 




1^4 






TABLE OF COIITENTS 

Section I. Introduction 

a. Mental and Physical lleasurement 

b. A G-eneral Statistical Theory 

c. A Suppostitioiis Example 

cl. Indexing; a Mental Characteristic 
Section 2. Tlie uroups and Data Studied 

a. The Groups Studied 

h. Tlie Tests Used 
Section 3. Test Administration and Scoriiif^ 

a. Primary and Secondar,- Index Variables 

b. The Difficulty Factor Eliniinated 

c. Methods of Adniini -storinr* the Tests 

d. Authors* Plans of Scoring 
Section 4. Methods Srrployed '^ 

Part I. AlIALYSIS OF THE INDEX VARIABLES BY \^iOLE SCALES 

A. Coniparative Validity and Reliability of the Indexes 
Section 5. Relationship- s between Index Variables on the 

Sams Scales 
Section 6, Relationships between Scales by the Same Index 

Variable 
Section 7. Tlie Reliability of a Scale by Different Index 

Variables 
Section 8. Correlations with Ap^.e 
Section i^. Correl3.tions with School Marks 
Section 1^. Su^Tiary of Reliability of Indexes for "hole Scales 



TT 



B. The Discriminative Capacity of the Indexes 
Section 11* Capacity of the Indexes to Discriminate between 

Individuals 
Section 1^. Capacity of the Indexes to Discriminate betr/een 

Groups 
Section lo. Practice Effect with Repetition 

Part II. ANALYSIS OP Tli^ INDEX YABIABLE3 BY CO.'FONMT TESTS 
Section 14, Inter correlations of Variables for the Otis 

CoRiTionents 
Section 15. Correlations of the Otis CoT'oments with A{^e 
Section 16. Tlie Applica.tion of Relip.bility Formulae to the 

Component Tests 
Section 17. Safiimry of Analysis by Coiiponents 

Part III. SCORING FOE.iaLAE 

Section 16. Tlie Linear Form, S - aR -f cW 

a, Forrrralae with Highest Validity 

b. Liimjsations in the Use of the Formula, Sraf^tc\Ar 
G. Use of t-ie ForuTula, S ~ H + CW 

Section 19. Sininle Ra.tios 

a. O^ie Correlation between Speed and Accuracy 

b. The Validity of Simle Ratios as Scorin^^ Indexes 

c. The Reliability of Ratios as Scorinp: Indexes 
Section '"''. G-enerai Suiirnary 

Appendix A. CorrelationsTables for Reliability Coefficients 
Appendix B. Sieorems Relatin^r^ to Correlations 
Bibliorqraphy 



p. 1 



Soction 1« Introduction 



, r- .(. i . . 



ifficultiss in mental -aeasurement are 
due to tlie fact that the method is necessarily indirect. 
This i::rclios Ta^raoness not only in the thrin- mm?•)^T(^Af but 
also 131 tho precision of Uio result* .jsay physical aeasure* 
ments on the other hand are direct^ raBkim possible a clear- 
©r definitian of the traits aeaeiu'er- nn-l --reater accuracy of 
their dctermnation. The caso of u ^oy to :jg woip-^ied and 
also tested for intelligence id.ll brin-r out this contrast* 

If the boy in r^laced'on thR -nhyf^ical seal© and weirded, 
the measiireiawnt iu recorded diroctiy in objectively dofinsd 
units J accuracy of deteniiination depending: upon the instru- 
:nent and the individual ^ymkinT %hf>. -ne-nTrrrrimt, 'T'ne reaction 
of tho joy hiiiiself at the ti^no of ii.io -iVGigiiin^ wiil have no 
effect on the rosult* In case a mental measurenient of intei- 
lif-'QncG in required, it in neces.-ar; to -ib-Tiit, t-^ the boy a 
seriea of questionn to t;hich he niust re3;;oua before a score 
for the msintal trait may dq obtained* This very indirect- 
ness of Qjrrryrofxfh I'^.'^r'';' /: '■^^■nbirniit.^'' re.-^ardinft; the trait 
measiuod, and to uncertainty in the precision of the score. 
Thus in passing from iihysical to inental measurement the role 
of tho boy has shxftof! fro ■ nr^Wn ;., active one, 

thoreDy coi:npiicatin<7: t.,Q uiioie prooecurc. 

It nrny be pointe; ' '. ;^rocediu:e in 

the c^n?" : ^r!r> nnr^cj^nnts has y^v .railed 



the phynical methocl. Unit3 liaTo bean defined and scales con- 
structed. These units for the most -^art aro functions of 
group Yuxitiuiiiby on pjix-uicaiax^ bypw.: " v.terial, ajid con- 
sequently lose in accuracy and simificance when applied in 
iiKiividuK.! measur cnont* Tii-? "irrtillelisiTi in mathod has even 
been carried "' " ' ~ *■" ritteuipi bo deterniine "zero points" 
for certain mental abilities so that " just not any" amounts 
of the traits could be used as reference -joints. In the C8,se 
of the ineasurofnont "/' :■',-- : v. - .araruioit ther- 

mometer does not moan "just no heat," but noYertholoss serves 
as a convenient reference T5ointo Si-iilrxly a r^ood many of the 
forraer nolmious 'zero r*oint" '-■ lontai /aeasurenient imve dise|> 
-pestxedt or have noved up to the median where they belonp:. 

b. A Cronoral St ^^tisti c al Theory 

The difficulties briefly sketched in the above r)ara- 

f^ra-nhB su'™"^ost the f^eslrabl''.it'r ef ?;T-ie -"rere rnnoral method 
of approacii to tiio problea of irioiix^i .uoasui'Gmc^nt , ana indeed 
such a ::iethod has been irnplicitly unal in some of the later 
work on tost reliabilit," rx]& validity- In order to Tit ideas, 

tlierefo.:. ^-^rtain terns will n-r^ ha deriued so that tnoir 

meanin.": will jq clear thraur)iout the saboequent discussion. 

Tlie term characteristics rdll be used to denote the nay- 
sical, :;ientai, or social traits •which i iidiYidrnds oi a rygui) 
have in coTi>non« The ^ou-p may consist of a nuiaber of persons 
or ttiin^s eac : ' -rhich Trcist nosner^f: the characteristic in 
question before any statistical ztm.'j is possible. A man of 



p. 3 
certain heL^-Vit, int0llieo;ence, and 7/ealtli fumislie-s an example 
of an individual with the three types of characteristics com- 
monly studied. 

Tlie pharjG of a characteristic ins.y be briefly described 
as the status of an individual rith respect to the character- 
istic. This conception is sn iiTiports,nt one because it is in- 
troduced for the pujrpose of distinf^ishinr: a particular thing 
from the number that niay be attached to it. Phases may be nu- 
merically or verbally expressed. Thus in describing the char- 
acteristics height and political affiliation of a certain man, 
the nur.ber 68 may be attached to the phase in height and the 
word "Republican" to the phase in politics! affiliation. It 
is conceivable that a nunerical scheme for the latter might 
be worked out, but the phases of political affiliation ver se 
would of course rerfiain unchanged. 

In order that a characteristic be n u/iGrically indexed ^ 
it is desirable that its phases be arranged in some order 
0.O-. iii,:e the points on a line. If the linear arrangement be, 
m8.de, the trait may be termed a lineg-g- ordinal characteristi., . 
For chej-act eristics such as heif^ht, an infinite nomber of 
phases arc assumed, indexed hj the real nusnber system (dense 
set). In the case of sucli chGracteri sties as size of school 
class, hoYrever, the nu-nber of phases is finite, and the in- 
dexino^ is fcco-T-ii^iihed by assi^ing only integers. The dis- 
tinction is essentially that between continuous and discrete 
series, the continuity and discreetness appearing; in phase. 

Finally an. index variable will be described as a quan- 



tity Ti5hose values are in one-to-one correspondence mth the 
phases of the character istics indorecl. Before proceeding 
farther, the nieaninf^ of these Tarioas terras will be illus- 
trated by means of an artificial exa^iple. 

c« A Su-^opostitious Exairnle 



Consider a set of 9^ cubes of horaoj^eneous material. The 
problem is to describe these cubes by ordinary statistical 
procedure. Assuiiiing that the size of the cube is the char- 
acteristic to be studied, some mode of indexing or index va- 
riable must be adooted. Takin-'*. the edge as a first choice, 
the distribution nmy be given as fGllo7;%s; 

edjte frequency 

3 30 

4 20 

The mean edge is clearly 3, ir/ith corresponding face area 9 
and volume 27 i.e. a= e";^ T-e\ 

A second node of indexing by the area of a face gives 
the distributions 



area 


frequency 


fa 


1 


10 


4 






9 


30 


270 


IC 


Z^ 


3T 


25 


10 


25^ 



The mean face area is nor; IO3 with n corresponding edo'Q 
and volune appi-ozirnc^tely 3.2 and . : i'espectively. 
Again indexing by volume gives: 



p. 5 



TTolianie 
1 


frequency 


fv 
10 


8 


?S 


160 


27 


30 


Cl^ 


64 


20 


1S8^ 


115 


10 


125^ 



The mean TOlurae is 39 9 with the corresponding edge and area 
approxiamtely 3.4 and 11.5. 'Biese results "aey bo set forth 
in sii'T-iary forra as followss 

TABLE 1.* IMl^S AMD COBRESPONDriG VALUES FOR CUBES II-IDIXED 6Y 

EmE. ilSA, A^S YOLOLIE ^ „ 

„ .,M ,1, 1 , 111 „ „ TI ■ . i ui. i. ri I — i-i T iniifc i ■ , ii jt- ■.- ^ ■HJLi M. ..^.^> j *- v :;;--»«'J^:-.--^--"'^- r ''- '^ ^ "■"■' ■■ J'- ■■■■•■ — ■ : '-■ — ^ . — i i> ^Myy^iM-M^ ™ . ■■■■^■'■■■P*! i^i^ 



Mode of Indexinf; 

or 
Index Tariable 



Area 

Yolnne 



Mean end Corresponding Indexes 



mf'',e 



tJm U 



3.4 



Arec 



10,3 
11.5 



Yolujiie 



^7 



33.2 
39 



InsiDection of these figures rcverls a complete incon- 
sistency imder the three raodes of indexing. Moreover the 
three distributions are quite different. Tho frequency poly- 
gon according to 9d.";e is syrmrietrical, r/hile the distribu- 
tions for area and volume are skewed toward the sraaller val- 
ues of the variable. It is also clear that the cubes re- 
mained ivt the same phases in pie characteristic size but 
that various modes of indexing' o-ave inconsistent results. The 
above example then illustrates the fact that althoup-^ a set 
of thinrrs be unaltered in phase, the for.n of the distribution 
and the statistical constants depend upon the particular in- 
dex variable employed. 



p» 6 



Fip;ure !•- Distribution of Cubes by Edr^e 



li 



Zt> 



iS 



2.S 



IS 



Figur 



- Distribution of 
Cubes by ibrea 



Figui'e 3." Distribution of 
Cubes by Tolurne 



It JTsay be well to point out briefly that the inconsist- 
encies in the aboTe example are clue to the relationships be- 
tween the index variables ;i*e. 



It mil be. sufficients to note here that simlar ineerxsisten- 

cies wil' -r?"!- iraienever the reiationsl>i;' between the index 
variables is other than linear (y=:ax+b). In the latter case 
the statistical constents will be merely affected by propor- 



tionality factors. Thus r.!hen height ir, -neaniired in inches 
and again in centimeters, the distributions will bo similar 
and all constants easily conTerted. by the sirm:;le linear re- 
lation s,hip, 

1 iiiCli ^ 2.54+ centimeters 

d. Inderlnp- a Mental Characteristic 

In mental measureraent the method is to set a body of 
material before a child end elicit certain responses from 
him. Tliese res^Donses are then recorded and combined in ¥a- 
rious ways to produce livhat is Icnomi as a score. Lloreover 
these responses exhibit considerable variety under different 
modes of administering the tesbs. Two boys took the Terman 
(Jroup Test of Mental Ability v^iich is administered under the 
plan of keeping the time constant. Tlieir responses may be 
set forth briefly as follows? 

Score (Autlior) Bidit Attempted ?/rong Accuracy 

John ISn 1^5 175 50 y71 

Henry 15*^ 14'' 15^ l'^ .93 

The problem of indexing here is^ unlike that for the c^se of 

the cube»s. Intelliri!;enco is the characteristic to be indexed, 

and this ir. possible by usin^f; the Rif?:hts, '.''rongs. Attempts, 

or so::iG com^jination of tliese ind^:>:: •^-^iri^.blo^ in the form of 

a score. It ¥dll be shO'.m later that for the particular scale 

in question the author's score is a rather Gom"plicated f^jcie- 

tion of t'lr Y-]'lables Ri'-^-'t anc ^'r-yix^:, :jl.o for the --resent it 

is sufficient to note the possibility of such a daaracteristic 



p. 8 
bein:<7; indexed in several 7/ays. 

Questions irrrnediately ailse as to the best mode of 
indexing". Ig it beet to ubg only one of the index var- 
iables available, or to co.ibiae several of tiiem,and if more 
than one variable is used how is the co.nbination to be det- 
ermined ? These questions are of most vital importance in 
mental 'neasui-eriientj arising; in one form or another irhenever 
a new test is devised ^:''-r\ ot'^ridardiEed. It may be pointed 
out at once that this stuay does not attem:pt any general 
solutions for these problems, but by analyzing a definite 
type of test material, aims to carry t'.:^ :'. .ivestio-ation a 
little farther than is possible under incidental treat- 
ment in the construction of a particular scale. 

Intellir':ence test ?imterial . a;^ uhosen for tv/o reasons. 
First, a lar^^e body of such data ?7as available. Dr. F.3. 
Breed and 'tr. E.R. Breslieh made available their escellent 
data for t'lree intelli-'rence tests Tiven at two parade levels 
in Tlie University of ChiGa{T;o Hi,p;h School, .jlr. Guy Capps 
also turned over two thousand copies of the Terman Oroup 
Test of Mental Ability, Forms A and B, -,■:.. ......listered and 

repeated in a number of Kansas hirh schools. A second 
reason for usin<r intelligence tests ijjithis study i^as 
that tho statistical constants i:-:^::OV:-r- from such mat- 
.erial arc more stable than from any similar test data of 
which the r-riter is arare. The violent a;id often inezplic- 



P.S. Breed and E.R. Breslieh, Intellio^ence Tests and the 
Classification of Pupils, School E6view,Yol.m( Jan. 1922)51-66 



p. 9 

able fluctuationn In p.nch conr,t?mts as correlation ooeffi- 
ciento caid. staiiuora deviations for ratmy or the woll~knoi;m 

achleTenient tests foi- .. .atics would only 

O'T'^liC'tf^ fni-*,''''?^''' : ' " 'tiidy of tills 

kinti«.->uc:i :;tati;^tic;al ' ility? '' ."eved, i!3 due for 

the most -art to tho lon-^h of the to ts and to their care- 
ful Gonf,t:aictr' „ 

Sectio-ti ^ Groups and Bats. 
a, Tho G-ro ^ pi StucUecI 

For nn rnal^rfii'cnl f,t/Uf'?y !*if this ty^e it i" de?!ir?-blD that 
tlie groapii stusjied siiouid possesa two i. .■; ortant properties* 
They should be l&ic^Q enoiit^i to insiu'-e reasomible stability 
in the ^tr^.fAntior)! co^^atpnts; ., :-nca"'id, they nhould be 
as hoiaonioiiooaii as 7303sibio# iiopro3feiioo.tiY& s£L;ipiGs froia 
lar'^e '^crnulations arc ©xbreniely difficult to obtain in test 

nciTk ''rr " • ' . -■'i<^T^^^ v.^t ai^rtnntialf to tho 

prosont ,::v::iu.y» ..GvorUioloG:; for UiC iarnest gi'oup described 
belo'- . .ith sar.'iplo of 135 out of ca:;QS wae t^iken by 

co-nb-^ ,. '-bri^^ ^r' rrfth r--n"To-rf.-iatGly tv> :■; nean 

scores as tne t>ovai i;^roi4p» 

iho p^Qiiv desir^eted horea" 'rt^Ae 7 couBisted of 

GrouTJo i iii'-^:: A oxi. inciuju-ed d^' }5Upiis re- 

speetivel\\fro!n the University of QhiGtir-o llim School. The Vth 
Q'rade . . '. " .■.■;.., c-nd ^©re pr©- 

pared Lj enter ^ie r6'f<:ui&.r first, yocu^ ciac -©s the follo^ria-^ 
year, there bein,n: n<| ei^iith grade In tlie laboratory Schools, 



p. 10 
Group I Hi.?Tji B diffei-ed from I Hio-h only by the addition 
of 10 pupils in certain of the tabulations. IMle the 
groups C^^'^rih(-d ^re imdoubt--^;!:' ?:olect.bh6y are unusually 
horao{>;eneous as ref^ards social status, training, emerience 
with tests, and ap:e. The j-rp;est f^roup,! Eirji C, consisted 
of 135 pupilG from the Ro 11a , Salem, and 61, James public 
high schools of Kansas, selected as described above. 

The age distributions for these .":rom?s .are.:»iven in 
Table '3. The '>--■- hi-: school pupils -..ere a year older 
than those in The University lii.rrh School, -mile the latter 
were about a year and a half older than tlie 7th <?ra.de ,t^roup. 
Thp '^.:.^:t:rVy:':.ioy^" in r^ch case present ^. fair defT:ree of 
symietrysthe standard deviations increasin,?^ with the size 
of the fi^roup, 

^^SI^L^LziMJM^^^'^^^'^^" "OK THE C-EOy?S STaDIED 



Age 


Crade 7 


I Hi.di A 


I liifTh B 


I Hi:0-)i C 


li^. 0-19.5 
18.5-13.9 
18.0-18.4 
17.5-17,^ 
17.0-17,4 

16.5-16, i;J 

16.0-16.4 
15,5-15. y 

15.0-15.4 
14.5-14.9 
14.0-14.4 
13.5-io.i3 
13.0-13.4 
12.5-1^.9 
1".0-1?.4 
11.5-11,9 
11.0-11,4 


■ ^ 

6 

16 

10 

8 

1 


• • 

* » 

« * 

z 

6 

9 
11 
10 

4 

2 

• a 

• 4 


o 
■\ 
i. 

» • 

I- 

•9 
■'J 

o 

2 

? 
10 
13 
12 

5 

3 

2 

* • 

• 


1 

1 

2 
6 

8 
13 
13 
17 

18 
21 

16 
J 
6 

3 

1 


Total 


50 


50 


60 


135 


Hean 


1^.6 




1 ,- y 


15.4 


S.D. 


9 '•■,' 


1.0 


1 • i. 


1.4 



p. 11 
b. The Tests Used 

mmmmmmmimm$m 'm * " iimi m i ■»iiii i m * m il— i— » 

Tliree intelligence tests were administered to the 
Laboratory School .-rroups by !lr. Breslich durinir, the year 
192<^-21. The scales used were the Otis G-roup Intelligence 
Scale, Form A, the Terman Group Test of Mental Ability, 
Form B, and the (Jaicago Intelligence Scale, Form B. These 
tests w®re all carefully scored by the Uioriter for Attempts, 
Riin'htSg ^^s^. Wrongs, and Accuracy, as rell as for the auth- 
or's score. 

The data from the Kansas schools consisted of the 
Terraan Group Test of ^Mental Ability Forni B, and the same 
tests Form B given the next day. From this group, there- 
fore, it was possible to obtain the reliability coeffici- 
ents. The large amount of labor involTed in scoring each 
paper for author's score, total Attempts, Eights, Errors, 
and Accuracy, and cajrefully checking all of the v/ork is 
largely responsible for limiting the samjole to 135 cases 
i^feen over l^^O ?rere available. 

In addition to the above data, school marks were ob- 
tained for 39 pupils in Grade 7. Yearly grades Tjere ob- 
tained for llatheimtics, English, and History. These marks 
were converted into a rou^ seal© dividing pupils into 
seven catagories for purposes of correlation. It is the 
belief of the xTriter that this degree of accuracy in treat- 
ment is about all such data warrent, inasmuch as these 
marks are intended to give only a rough estimate of the 



p. 12 
achievsment in the various subjects. 

Section 3 Test Administration and Scoring 

il PrinK).ry and Secondary In dex Variables 

In administering and scoring a test the following- 
variables moEt of necessity be taken into consideration 
directly or indirectly: Difficultyj Time, Atteropts, Eights, 
Wrongs J .and Omissions. This implies of course that the test 
material consists of a series of items T/^iere the responses 
;nay be scored right,i!?rong, or omitted. The six variables 
listed above will be referred to hereafter as Primary Index 
Variables and any function involving more thfin one of them. 
as secondary index variables. Thus if Accuracy be defined 
as Eights divided by Attempts, such a score would be termed 
a Secondary Index Variable. Also in order to save space 
these variables will usually be denoted by the initial let- 
ter of each \TOrdf i • Q. » 

D - Difficulty 
A - Attenmts 
T - Time ^ 
E -- Ei^ts 

W -= Wrongs 
- Omissions 

b. 'Tlie Diffi culty Factor El iminated 

The problem of the difficulty of the various items in 
the tests is one which must be settled before proceeding 
farther. This problem is i-m licit ly solved by the authors 
of the tests #ierein each item is given an equal or point 



p. 13 
value with all others. Tno questions arise in this cormec- 
tionj are the itenis of equal difficulty, and if not should 
they be weighted to obtain an accurate score! It is a Y-rell 
knowi principle in the theory of index nu^fibers that the 
longer the series the less the effect of differences in the 
welf?^its of the individual items. Test scores are, of course, 
really index numbers. It may be readily conceded therefore, 
that the authors of intelligence scales consistin^-:^ of so 
lonp; a series of itonis, are quite justified in assigning 
equal T/ei:<:^ts to each ite:-i rer^ardleGG of the better values 
which mir^ht be assigned on theoretical grounds » By way of 
justifyin^f^ this assuniption, sn example T/ill be given of a 
short series iTith considerable variation in weight from iteia 
to ite-'ii. If differences in weigiit are not significant in so 
short a series, they xTill be even less so for a very long one. 
This method is that of the unfavorable case. 

Test 2 of the Terraan Group Intelligence Test consists 
of 11 items the response to each bein^ a best answer appro- 
priately checked. One himdred pupils ■mere selected wiio had 
taken both Foitns A and B a day apart. The test papers for 
Form A were then scored for errors and rouTh weigLits assigned 
in the usual way assuming a normal distribution of diffi- 
culty. 



TABLE 3. -PER CENT FAILIria AND WEIGHTS FOR TEix^^iAH TEST 2 



Ite!-ii 


1 


2 


3 


4 


5 


6 


7 


S 


Kg 


, ^r, 


11 


Per C9n|5 
Failing 


8 


19 


29 


30 


26 

i 


24 


16 


■'4V 


65 


33 


56 


Wei,^-^it 


'-> 


3 


i 4 


4 


4, 


4 


3 


5 


7 


4 


5 



p. 14 
Next by esceedinnily tedious coinputation a Tjeir?;hted and 
an unweirfitod score for each pupil was obtained. Thus a boy 
responding correctly to iteins i, 3, 7, 9, and 11 on the test 
receiTed a weighted score of 21 and an wiweipiited score of 5, , 
the nuiTiber rip^/it. A correlation table was then nmde for 
weigj-ited and unweighted score ivith a resulting coefficient 
of A-uvAi - -^7 2 . The standard error in sstinatinp: unv/eigrited 
from 7/eir^hted score ,, So-.l .The correlation between the two 
miT/ei.p^hted forms of the test was then obtained, giving Aa6-.S"=i7 
The st.andard error Su in eotinrnting mvneirjited form A from 
unTjei;Q;hted B was 4.8. 

Thus the correlation between tf/o lor.os of the same test 
is much lor;er than that between weighted and unweighted 
scores on the same form. Also the standard error of esti late 
of unweip^ited from weighted is about one-fifth that from 
form B. The wei^ting of the items then gives a degree #f re- 
finement considerably beyond the roll -bility of the test it- 
self i.e. correlation of two forms. For 40 weighted and un- 
weighted items of different material the ?7riter obtained cor- 
relations of .9.^8, .9975 "^^S ■'^^ith the corresponding reliabil- 
ity coefficients between .85 and .9^. All of these results 
point to the conclusion tiiat for a fairly long series weight- 
ing of the separate items is unnecessary. A conrolete solutior. 
of the problem would involve experimentation with series of 
variouB lengths, items of vai^^ious difficulty, and populations 
of different size. Such exhaustive treatment is clearly be- 
yond the scope of this study. It may be finally pointed out 
that a nuTiber of recent achievement tests have appeared first 
with weiglited items and later with weights dropped when it 



p. 15 
was realized Ttiat a sli^it difference thes© made in the 
resulting scores. 

C» Methods of Administerinfy the Tests 

With the factor of difficulty eliminated the scoring 
or indexing problem is greatly siiaplified. The plan may now 
be described as the method of Unit Eesponses, the response 
to each item being scored as a unit point. Furthermore if 
Amissions be neglected or counted as errors ^ the reiriaining 
primary index Tariables are reduced to Time, Attempts, 
JRights, and Wrongs with the relationship, 

A =.R+W 
if omissions be coimted as errors* This assur/iption will be 
made in a subsequent discussion. The nujiiber of omissions 
occuring in the tests used was negligible. It thus a pears 
that four' variables 9 T,A»E» arid W ?/ill have to be'studied? 
the last three not being independent* Also all scaring ^for- 
mulae or secondary index variables will be functions of 
tliese four primary indexes* 

Two plans for administering such tents are possible. 
One plan is to fix the number of Attempts allomng Time, 
Rights J and Errors to Tary, Tiiile the second method is to 
fix th© Time, allowing Attempts, Rights ^ and Wrongs to va- 
ry. In the last analysis then, only tliree index variables 
need to be considered, the fourth being arbitrarilly fixed 
by the plan of administering the tests. According to the 
first scheme outlined, one allows all of the children to 



p. 16 
finish the test thus keepin,^ AtbeiTipts constant. The time 
is then record od by stop ivatohes or clock device and 
RifT-jits and Wrongs obtained from the pa.pers. By the second 
plan a fixed time limit is set for all the pumls, Attemnts 
Ri/3^t-^, and Wronr':s bein/^ then scored on uhe test papers. 
Th.e second method is obviously sinipler than the first, and 
is no?/ followed in the p^reat nmjority of tests of all kinds. 
Certain te^ts, it is truSB, neglect actual time but these 
are not considered here. The material used in this study is 
all adjiiinistered under the plan of fixin-^ time giving the 
three priiimry index variables A,Rj and \^. 

d. Authors' Plans of Scoring 

For the three intellir^once tests described above, the 
authors have set forth scoring foriiiulae for each test of 
the battery. These fornsjilae 5xe obviously expressed as 
functions of the primary variables A,xl, and I, while relat- 
ionship A - R-ff makes it possible to set down the equations 
in terms of any tvro of the variables. In the follomng tab- 
ulsjT scheme, therefore. R and W have been e.Tiployed throu";h- 
out» 



TABLE 4.- AUTHORS* SCORING FORIIUUE ON THE THREE SCALES 



Scale 










Test 
















r^ 




4 


5 




7 


8 


9 


10 


Otis 

Tearman 
Chica'"o 


R 
R 


■ 

R 
2R 


R-W 
R-W 


R 
R 

R 


R 
2R 

^R 


R 
R-W 


R 
R 


R 
R-W 


R 
R 


R 

2R 



p. 17 
It will be soon frora tabic 4 that ti70 of the scalcc 
consist of ten tests each waile the third inoludes only 
five. Considerable variation in the scorinp; formulae is 
also to bo notedo The otis Scale has 9 tests scored by R 
and only one by H-f ; Temanj on the other hand, scores ■ 
four tests by R, 3 by 2R, and 3 by B-W. For the Chicago 
Scale t:-o ^er fo'-^^"'^e a-^ear^ H- i W arid 2(R« ^f). All 
of the above foruiulae are clearly special instaaices of the 
general linear form 

where a and b are constants* 

The simr-licity of the Otis' scoring formulae imnedi- 
ately raise^ the quostion rs to the advisability of scoring 
the other scales by the same method. This problem is dis- 
cussed in the following sections. Another question concerns 
the for:Ui..3 R-W. and R- ^W. These f ornialae are employed for 
material of the True-False t^?pe and t}.irer3-choice variety, 
on the supposition that"' they correct for the element of 
guessing for 3uch tests. Tliis proble-^. "' ■! also be taken 
up in some detail in later discussion. Finally the doub- 
ling of scores for individual tests is a point that needs 
consideration. So far a3 the i?vTit9r it abls to determine, 
this doublin'-^ T?as effected because the author of the test 
felt 3uch tests to be worth about twice as much as others, 
or wanted to increase the total points • o'^'-"'JIo to soiTie 
convenient number. The Doint at issue is the same as that 



p. 18 

between Fei'^j:i,tedand iinweiHited. items* A .crrs^uate student 
ma,de a ^y^ii..^/ of the Cliicago Scale regarding'; this problem 
and found that the welghtin,g of the tliree tests as indi- 
cated in Table 4 affected the correlation bat slijp;htly, 
the coefficient between ?/eip'J:ited -iind imweLo-jated scores be- 
ing .SB. 

It thus appears that Ei.f^its are the basis of most of 
the SGorinfr fonmlae employed on these tests and thet other 
forms have been used to correct Right responses for gaess- 
ing or to wei.^th theT±iole test because of the relative im- 
portance in the battery making up the scale. 

Section 4 Methods Employed 

Taci -^--oneral method of this study will first be to an- 
alyze the interrelationships of the index variables invol- 
ved, and tlien to set up certain criteria of good indexes 
and attairnt to evaluate the variables in terms of these. 
Obviously this As an indirect approach and it must neees- 
saril^y be so from the nature of the problan* The actual 
technique employed will involvo a considerable amount of 
correlation, for this is the best method of studying the 
relationships between index variables fro i tests. For the 
case of the cubes deserib^^''"''. «bova the index variables 



p. 19 
were functiondly related,!, e. 

Tolume - (edge) 
Such f'uiictionality can only be approached \)}[ empirical data, 
the correlation coefficient f^iying the most convenient ap- 
proxii?iitio:;i for linear fimctions. Thus if score iaid Rigiit^:. 
are coi^related to the extent of .98 icLth regression lineai% 
there is a very close approximation to the functional re. 
lationship S - KS. Tiw chief adTantag.. o: correlation is 
that it gi7es a nuiTierieal estimate of the closenesG of such 
relationships or approach to linear fimctionality. 

Index variables mil be analyzed by batteries and by 
single tests, ige, school rii3xks, and otlier intelligence 
tests will be used as criteria against ^iiich to check the 



var i ou" f or': liiil ae . 



All of the calculations beloF have been preformed by 
the Tnriter and have been checked T/ith care. Correlations 
were obtained by the usual product L-ionient method with a 
specially designed correlation forra. Tliis was found to cut 
do?fli the labor of calculation very rmterially especially 
when "batteries" of coefficients 7/ere required. Blakenian's 
test for linearity was applied in a few cases with the re- 
sult that the miter believes the great i:ia,jority of the 
tables Gzliibited sufficient linearity so a^ not to reduce 
the correlation beyond the limits of probable error. 

PaxL I Tfill be concerned -Ith an analysis of the five 
index variables by ?iiole scales. By the indirect method of 
correlation, the relative merit of the indexes ?dll be de- 
termined. Part II will involve a similar analysis of the 
individual tests of the scales. In Part III certain form- 



ulas ?dll bo develoriod end their validity fmd reliability 
detGrmined in '^"■f'J^i"^'^.'^tical ter-^f^*. 

Part X iinalyais of the Index Yailabl es by Wh ole Scales 

A>~ ComnaratiTe Talid^ t y and RQllability of the Indaxes 

Tae most direct approach to the problem of indoxini^, 

will '^^' ,y '^■''^T''^ ^'''^ '^■'i^.'i.-Of* -^^ -'^l'^^: of p.nri~**;ir' js^ Q2« 

hibited in i!able 4 with the slm^'lar plans of coimtinr; 

meroly Atts^w^'ts- Hi'-^^t^ir'Srrorn, and determining Accuracy 
for ^t".', ■:-:->nHi^iirr>' -'■?i Vyr^iTA-i f^fi rslatiTS 

merit In reiiaDiiity of tiieso siirrf\lGr primary variables 
#ien soTsral teets are poolod to ,r?ive a scale score. The 
torf, "f'C""^'"' '^■f^'^''"^^ "J f^ hn-jTi «?i*T,-"' A',*'' ,15. '^.t-"^^'>«^'^ ^h it from 
test soor© i^iiiiosii will be used to aonote Uw score on on© 
of the eonrnononts iTkhiri up the tot^l scale* IPhus the Otis 
Scale i:. irndQ w^ 0^ '^^ ,-.,-,' 7-.-.r-nn^-if, ■*, - . ^ 

Fiv tirpoB of eomrjai'ifion will be ;:ii>jde in evalu0.ting 
the indos: vari-^blars for rm??ral reli"-bility» Thee© includes 
ft. B©latlona : ■ /iju-vr;;ai index Taria •. ^^..0 smse scale 

b* RelatioT-shipji; between scales by the sa.ne index variables 
Co Heliabillty of a scale b|r different inde:-?; variables 
d* Correlatic-" "'^^'^ "^-'' 
©. Gorrolations ?7ith school ?narks 

Sec tion 5 Rel5^1onohir>n bet^oon Indo:: T-iri^-blGS on th e 

Uomi Scales 

As pointod out afoOTe the relationships between the va^ 



TD. 21 



ibles camiot in general be functional for empirical data 
such as thopiG obtained froa montal tests. Noyertheless iu 
irill be Taluablo to discoTor tlis closenGSii of linear re* 
lationship as indicated by the correlation coefficient. 
These results are set fori- '^abl^ 5, All of the corre- 
lations were copjoutsd on total Aotor.rotOj, Eiglits, eto. foi" 
the entire scales. The agreement froui sroup to group and 
scale to scalo betiveen correlations for the same two var^ 
tables is striking- and may be viei^ed mUi pardonable sat- 
isfaction by one #10 has viewed many isiexnli cable differ^ 
©nces for other t;/i39s of test*^- ■■■-'■/.'^ : _: . ,' .■; 
TABLE. §•- COEKEUTIOMS BSTWESI^I YMII^BI^S FOH THE SA'IE SCALES 



Scale and 

G-rom:- 



Pairs of ITariablas Correl at'i'-; 



S^A' S^R 



Otis 7 
Otis I B 
Ter*? 
Tor. I B 
Chi. 7 
Chi. I B 



S> 



+.72: "♦••99 -.52 
+.67!4'.99 -.59 
+.59 j-f . 96 : -.45 
+»&9 '•f.96i-.67 
+.45 +.9'^ -.54 
+.54 +.95! -.71 



S>-^ A>H 



-*»,73|+.73 
+.81: +.37 
+.06 1 +.7^ 



T 



R! 



A 



+•33 
+.69 
+,02 



+•74 
+.59 

+.62 



+.21 +.13 1 -.51 
+.41 -*i.6> -iS*^ 



+*14i+.10 
+.31 -.1^ 
+.19:+.^3 

!■..- 



-•61 
-.39 
-.05 



+.70 
+.77 

+.53 
+4 73 

+.59 

+.7D 



W>| 



-.91 
-.93 
-.94 
-.93 
-.93' 



Mean 



ff.Ov' 



+.56U.58 



+.76 



+.C9 






ori 






+.69 



-.93 



.- v^- 



for example the correlation between authors' i;icore and to- 
tal Attenrpts for Otis G-rade 7 is .72 +.05, i/hile for Otis 
I Hif^h B it is .67 +.05. By the forimila P.e<x^A-= \f^f+tP.t^) 
the difference between the two correlations may be written 
in the fonp. diff*-.o5i.o7 , indicating that a difference 
of .05 is unsignificant. To be si^dficant such a differ- 
ence would have to be 3 times its P.S. Sirailar coiiipari sons 
between groups show tijat nearly all diff- 



. -, « , p. 22 

ences may be accounted for by tha fluctuations in samp- 
ling. Moreover if tests made on th© same group with dif- 
ferent scales be considered, as independent santples, most 
of the inter-scale differences are also insigiif leant. 

'^Ii:. table yields some very interesting results. In 
the second coluam it ?all be obsenred that -the correlations 
between S and R are very hi^Ji, the meen of the six coeffi- 
oientn being +.96. The extremely high correlation of •99 
for the Otis scale is duo to the fact that only one of the 
tests in this scale is not scored directly by Ri.ghts. The 
above result, therefore, raises a question as to the ad- 
visability of scoring this lone test differently from the 
rest. Moreover the two correlations of .96 for the Terman 
Scale indicate that even mth a fairly coniplicated system 
of scoring, the agreement between autliors' score and to- 
tal Ri*ts is extremely close. With still more coiirpli Ga- 
ted forujulae in th© Chicago Scale the correlations bet^/een 
S and R are 8/^ain very high. For whole scales, thea^ a 
mere enumeration of the total number of ..:.; ;.xu res-^^onses 
gives a result very nearly proportional to that obtained 
by the use of various formulae for the individual tests 
makin.^ ir^ the scale. It is to be noted. h-'"-ey^rp that the 
above result aj^ears to be valid -when single tests are 
pooled to give a total score. For single tests, changes 
in the scoring foriiulae have marked effect upon the cor- 
relations mth criteria as will be shof/n later. 



p» 23 
Accuracy fumishos tho next» hi;'^iiest correlation with 
Score, tho meen coefficiont beinr; +»76j liiile A and W come 

iv 

next Tdth correlations of 4-. 59 and -.58 r^^r.-nRctiTely. If 
Score be adopted as a criterion, thereforo, tiio best indsx 
variables in order would be Pdghts, Accuracy, ejad Atteiirpts 
or Error.;;". It .'. ;■ -..:,tlier sun-'d j^inr^ fM.t nnrely counting 
the Atteiiipts or Errors on thc30 toots givos so good an in- 
dex of intelligence. 

In the last coluim of the table a very hir^h : .:.; con- 
si stent nep:ative correlation is found bettreen W and -^ 
This means that a pupil #io raakes a great •.mny errors is 
very inaccurate as measured by the indeic-^ , or that such 
an indez variable is an ezceodrngly good measure of what 
is ordinarilXy understood by Accuracy. It is of some in- 
terest to note that the above ..-^^w......;.!.,, becomes -.1,W> 

vhen the number of attenrpts is fized and the rels^tionshix? 

A-= R^ VJvT 

still obtains. In this case the functional relationship 

holds strictly, and this i-^-i^/--- --^r-Poc- ;:3;r^ativ© corre- 
lation as is shorn raore fully in a section below. 

If the index variable A be used as a measure of speed 
a number of interesting xelationshi"''" ".r-; bro^ifrjit ou,t by 
the renrdning coefficients. Speed and accuracy are evident- 
ly uncorr elated, the highest correlation between: these two 
variables being --,16 ^,.»r.9 which •" " '^isi^^nifieant because 
it is not even tivice its probable error in amount. Moreover 



p. 24 

the difforencGs in si^ are such as to giv.; x •isen corre- 
lation for the coliimn. of less than ,Ci» It will be furth- 
er noted that R in correlated mth A to the extent of -!-.69 
on th- "T^T^^ ^ ^''^ '" •"'^;'-i A f^i^e- ' ■■■•'■•r-.i eorrel^'.t-^-'-*-^ of 
+«24# Ml a? 6r age. Gorr elation of +.59 between S end A was 
preyiously noted. These results indicate that the way to 
get ^ hlff^i intellifrence ^'•-■-•■■^ -'/; ■' - :-ork fast. By working 
fast one is likely to xuake more rrd stakes, but he Is much 
more likely to ,n^et more itema ri,p^t and make a hi,c^her score 
than 5^f ' orkdv. .ior^ oxOvdy. Intellifr--"- f,^~\tn have 
frequently been called "alertness" te.cts. The above find~ 
inf^s indicate that with considerable appropriateness they 
might '^■''": ''^■^ termed ''spec '•'''' ^--^"t". ^h? -.n sumption thus 
far in deterrnininf?; general reliability has beea that the 
author3*score is the best index of intelligence. If, on 
the other hand, Accuracy had been assumied to be the best 
indexs the speed factor would have been eliminated, there 
being no s;enoral tendency as indicated by tlie zero corre- 
lation for a ijupii uo get a high or low accuracy score by 
changiFig his speed. It iiill also be recalled tliat Accur~ 
acy was hi.^hly correlated I'dth Score (■J-.76) so that f- 
as an inuw-c o-ariable is t £ar^^u#A y coasistent uith S witn- 
out being miduely influenced as is the latter Tariable by 
i^e undesirable speed factor. 

The average correlation of -.49 between R and W means 
that the raore items the pupil gets ri^it, the fewer he is 



T). 25 



likely to get wrong* If Attempts ara eonstojit tko above 
correlation becomes -i.^^ as in the case of Wrongs and 
Accuracy. F:'r/.lly the yrr^y^^ '>"-^r><^. T? and a -re found to 
correlate an the aTera^~;e ^oCj, a coefficiont viiidi becomes 
♦1*00 -^ea Attempts are constant. (Theorem 3 Appendix) 

Section 6. Relationship's betT/eon S cale s bj- the Same 

"fh© relationships between index Yejriables on the 
same scale have been disGossed in the precedinp, 'Bection, 

with uJ.O :. ; ■a'',-*- ■*"h^'^ ■'■-■^ r— ,virM--^T r,--/?: ' -r^-;i^■!^ -n th 

score as a criterion is Rights, Accuracy, sind Attempts 
or Errors. 15ie variable R r^ossesses a si>Trlicity #iich 
considered hi oonQectit. ■ ' v' '-' -■--,^;^ ---k:' 
3 suggests that it mglit well be substituted for tiiat va*- 
riable in inde:dnf? bj/ batteries of tests. The index x 
was found to be in close £^;;xuuaenu -atli S and R and to 
possess the advantage of being unaffected by speed. 

TtxG next procedure mil be to evaluate the index 
v-s'xiables indirectly by determining; the correlations be- 
tween pairs of scales indexed by the same variables* The 
closeness of thi? corresT^ondence id.ll f^iv; a measure of 
ti^ effectiveness of the particular uoc.: indexirig. The 
former comr/arlson was, inter ^variable, the present is inter- 
te-lt. Table 6 o-ive?^ these results for p-rn-i-r. I Hidi B and 
Grrado 7. Coin|.jarison of this table mUi Tauio 5 reveals the 
fact that the correlations in tlie former are rmich more ne- 
arly tlie same size. All of the coefficients are signifi- 
cant, T^ile there ar© few of the differences between any 



p. 26 

two Ttiich may not- be attributed to saanpling by the usual 
method* Ths neenn for the coliirmB and rows of the table 
brinf^ out the stability more clearly inasmuch a.3 the de~ 
viationa frohi these are in ;:i03t caoes Yary sliglitt 

« 

The really surprising; results exhibited by this table 
are indicated in the col-oirin of means for tlie various in- 
dexes. Here it appears that with inter-seale correlation 

as a criterion, S,E,E9 and 1 are all about equally good 

I 

for purposes of indexin.^, ftiile A is someviiat poorer than 

the root. To discoTer ai.iiost as hif^i a correlation between 
two intelligence scales by merely tabulatii^ total errors 
as by usins; the author's score or total Rights is at first 
somev;hat surrprising. It is a closer a^eeimnt than migiit 
be expected from the areraf^e correlation of -*58 between 
S and ?/ from Table 5, Inter-test correlation giTOs a meas- 
ure of the extent to i-iiich two tests a^ee in msasuiln.?^ 
the sacie character is tic. From Table 6 it appears that this 
a^;r©em9nt is about equpjly close isi-ien any of the four' in- 
dex Tariables is em^oloyed^ the general ordar of merit be- 

inp, indicated in the table as S,E,IljW, end A« This order j 

I 
it will bs recalled 3 is in harmony with thtit found in the 

preceding section. It may finally be pointed out that the 

lack of variability ainong the coefficients stron^c^ly sug- 

fi!;ests that combining these index Tariables as in the case 

of S f/ill give but slightly better index when batteries of 

tests sudi as these are employed. 



P* 27 

TABLE 6.«IIJTSR-SCALE CORRSUTIONS BY FIYE IWKi YAHIABLSS 



Judsy 


Pairs of Scale,-} ajid G-roujos 


Taxiable 


Ter;:i. a Otis f''?9nn,)cChlc. GhioicOtis 


Mean 


Order 




11. 


H.S. 


El. 


H.S. El« 


H.S. 




Score 

ktiermtz 

Wrono; 

Accuracy 


+. 7,y 

+.73 
+.72 

+.70 


+ .63 
+.72 
+.82 
+.63 

+.88 


+.66 
+.5C 

+.59 
+,6<^ 

+.61 


+.76 
+.46 
+.71 
+.62 
+.77 


+.78 

+.78 
+.66 


+.79 
+.53 
+.83 
+.65 
+.74 


+.76 
+.56 
+.74 
+.69 
+.73 


1 

5 
Z 
4 
3 


M®an 


+.71 


+.82 


+.60 


+.68 


+.67 +.71 +.70 





Section 7 Ttid IleliabiIi^^.:/ of €. Scale u:; Mffsrent Indj 

Variables 



Another method of studying the relatiTe merit of the 
different index variables is to obtain the reliability 
coefficients for a scale under each of the indexes. Q-roup 
I Hi^ C was iised for this puiisose* It mil be recalled 
tliat tliis group consisted of 135 pupils idio took Form B 
of the TeiTdan Scale and Form A of \..,..w ..a...:., oest on the 
follo?ang day. The reliability coefficients in this case 
are given by the correlations between tlie vaxiables on the 
t!^o foruis of the scale. ThsiJ^ o.^iTGlation.j .ix^ given in 
Table 7 Ydth the contingency tables presented in the appen-. 
dix. 



p* 28 



TABLE 7.- RELIABILITY GOEFFICISITS FOR 1?EBHA:^ T^oRrfS 
A AI© B Oil G-IIOIIP I HIGH C 



Tariables 



Ssors 

Att,Gr-Rts 

Accuracy 



EeliaMlit- Goef.rieient 



+.684 ?.C3;^ 
+.896 +.C11 
+.737 TMi 



It is at ones apparent \2u^'^ ' ■ oeoia na^ 

the hicrhsst reliability mth a coefficient of .908. A 
glanc , t'T-? corr^B-'^-.indin.T correlation table in th© ap- 
pendix wiii. give tiiio ro^uit aore liiyuim^i. Here -the lin- 
earity of regression is at once aopai'entj and tlrie rema^rk- 
able a^eesient 9siiibit©d In graphical fora. Close corrs*- 
spondence of this typa appears to th© m^ltsr as one of th© 
most Bi^iifieajit acliieTements of S'tai-idardized tests. Just v 
irtttiat U... to./ ^-B^^ws:.. ^.,: indfx lo „rix. ainbigaouSj but 

to index any mental oLaraoteristia mih saoh a high degree 
of reliability is in itself a most not9^?orthy achievemont. 
It duuaiu, ■;. - o^-.Li.j in mind '&t all such reliability 
coefficients (and indeed all such eorralations) depend up- 
on the f^roup. Selection mil in general tend to reduce 
such correlations iMle heterogeneity due to such factors 
as age mil tend to increase it. 

Retumin.'T to th© reumining coefficients in the above 

table jif© iind that the order of reliability for the five 

variables i^^ S,H,R,Wj and A, This is precisely the order 

I 
obtained by the method of inter-t©st correlation as shOTOi 

in Table 6. 



p. 29 
If the symbol /Ixx be employed to denote the re- 
liability coefficient for a test indexed by the variable 
Xj thQ differences between the correlations in Table 7 
may be exhibited as followsj 

J^« - A.AA ~ -^.224 +.034 difference is simificant 

A^s _ /xk^ - 4.,012 T,oi5 difference is insipjiif leant 

/Lss - A».w ^ -i-All +.^29 difference is significant 

/i-.s -~ rt^L. ^ +,067 +.020 difference is significant 

fhe forniiila used for calculating the probable errors of 

the differences is, 

The sinali differences between the reliability coefficients 
for S and R is insignificant^ T^hile the remaining differ- 
ences are sufficiently large in comparison with their prob- 
able errors to be significant. Thus from the standpoint of 
reliability the Terraan Scale is indexed equally well by 
author's score or total riglits, and next best by accuracy. 
A reliability coefficient of .74 is generally considered 
highs and it is remarkable that total errors on the two 
forms should correlate to such an extent. The difference 
between A-sb and Ivuw ^ howeTer, is over fiTe times its 
probable error so that the reliability of errors as an 
index is significantly less than for score. Similarly 
Attempts furnish a niuch less reliable index than Score , 
Rights J or Accuracy. 

. In this part of the study no attempt is made to an- 
alyse the indiridual tests making up the scales. N®Ter- 
theless it is interesting to note from Table 4 that three 



of the ferriCT, tonts r»ro scorsd R^Wg thrn'* "^ -^ad four 
Ijy I aXoRO. The uiiicrGiico iu tlio roii^^i-ity oo©ffi~ 
cimt P-ss wider tliin plans and /I^r bjf Qoimtinr:: more* 
ly ri^^itn in '^'»^V.^i,-H5 aj^ nhoTm above. ' Formulae of trie 
type M oua iv-w, -i^uc appua^- iu ha?o no effect on tao 
rollaMlity of the total ;30oro, and tlasir us© for sue . 
scales i3 open to quQsticm* 

Section (^rreXaticr.^ wjtlrik^ 

^10 age factor is al?®y3 of interest ihrn fitucly* 
in^ tost results* Table 2 vitidi (^wqb the cire distri.- 
butlons in half-yoax inteiralSj shoT/s raa\^]oa fox- 'Uie v..** 
rious ©roupEJ frc^i four to ®l|§;it' years* la IJable 8» the 
corr^V'^'.t'i^'^'^Ti.'"- botrro^'^}'; ■'^.'"'r fflnsl. thn four index variablnn 
are given for Urade * ana i iilg. ii« A consistency in tiiose 
coefficients is at ono© apparmt« 'Bib laosn correlation 
of -«4i between a'!;o and score indioatos that the younr:* 
or pupils ar© brif^Iiter tlian tlio oldar mios within Ui^ 
B&rm g^ade groups Siii3il8xXy Uib tmm. correlation^ *»389 
**299-*26, and *^2^ slio?^ tlmt -^...o yourjger pupils «^t 
raore itonis ri/^it, are E3or$ aeourat©» aro sp©Mier,» asad 
mak© fouor errors tliaai tlie older ciiildren* 



p. 31 
fmm 8.- GOEKBMTIOHS BITWSEI^ AGE MW INDEX TAHIABLES 



Scale and 


Ap;@ with 


Group 


Score 


Attempts 


Sigiits Wron^^s 


Accuracy 


Otis 7 
Otis IB 
f erman 7 

Termn IB 
ChiGa,«;o 7 
Chicago IB 


-.41 
-.40 

-.47 
-.39 

-.39 
**39 


-.37 
-.27 

-.22 
-.21 

-.21 
-.30 


-.37 

-.39 
-.40 

-.37 

-.05 


+.06 

+.27 
+.17 

+.28 

+.20 

+.24 


-.24 
-.32 
-.37 
-.34 
-.33 
-.26 


Mean I-.41 


— •??6 


-.33 


+.20 


-.29 



The superiority of the yoianger child is eTident, therefore, 



no matter which indez is ©raployev 



^.1,, 



"DesuS* 



When arranged according to the size of, the correla- 
tion ^d.th a^e the order of the index yariables a-nr^ears from 
the means as, S,R,B,Ag and 1. The three variable^i Score, 
Eights, and Accuracy retain ttie order found in the previous 
sections. MoreoTer the mean correlations for score and 
ri^ts with age ai"e vei^y nearly the same, so that the su* 
periority of S OTer R as an index is again slight if any. 
The usual sanrlJn.<^ forrraila reveals no difference in the 
correlations ttiat is of sta.tistical sigLiificance. 

Similar correlations are given in Table 9 for the 
lajT^onf ., coefficients, though someitiat smaller 
are in hanflony with those of the preceding table. The de- 
crease in size is probably due to the greater range in 
a.ge. 



p. 32 



!?ABLE 9.- GORREUTIONS OF INDEX TJJIIABLBS WITH AGl fui 

TSKIAII SCALE FORM B 



G-rou-o 



Age Tfith 



I M# B 

I Higi C 



Score Atteaipts 






-•16?.<^6 



-,X4;f«<^6 



Eights I Wrongs 



Accuracy 



-.37t.O© i -^.28^.09 



*.34??.08 



By increasing the range thro-o^aseTeral grades the corre- 
lation becoraaa roBitiTe. Bi fact heterogeneity, or lack o: 
selection appears to have a ci^rious effect on correlations 
with a^e« For children of exactly the sain© age the coeffi- 
cient i.. ..: 3oi3rs0 zero; lih.fm. the i-ange is increased to 
several years as in the typical grade group , the eorrela* 
tion is negative. As the range in age increases^ the co- 
efficient approaches zero a^-^ain, and finally ^.icisses through. 
this value to positive values of considerable size if sev- 
eral grades are pooled to give a long range* The theoret-^ 
ical curve for tlie eorrelatior. .x, efficient will thus have 
an appearance resembling that in Figure 4. 15ie negative 
correlation for Uiq age interval OA has been accounted for 
by the appearance of older retarded children in grad© groups* 
%i3 explanation, while plausible, does not seem satisfac- 
tory for groups such as G-rade 7^ ^nich irj usually free from 
chiMren of tliis type, Tiie positive correlation increases 
from A as the ag© span is lengthened. 



p» 33 




Fl^» 4 'Sa0o.r9lji0al curre for eorrelatloa iTitli om 



■i^^^'^*-"i^-^ -r ,.,i;'.jyrC.4£:..C....3Tl3 ^771'GJI 50X00., 



.. Ic:r!k:o 

i>|pi||ii||| ii ri illl ^ iil l) 



wiiiio school msKB are obviously in;-ccurata est! 

of the ability or adiieveawiit of -upiiB,. nevertheless th©y 

nK!.y sorre ':',?^- "■ useful '"e'''';'o« '■>'"■-'•- *.'-►. ^.^- -■•■•■. ^ -•% tftst 

result.:;* yiioir rolatl?© iii«cc'ajrac^? isuisi beeii gi^ossly ©xa^- 

a b 

g@rat0(U Burt, Proctor, Kelly, mid othears have shorn tMt 

the predictlYO value of ....w... rnarks ...:. v..wu.i as hic 



p. 34 
that from intellip;enc:v .„ achievement tests. MoreoTer in 
the prssent ooiiiriarisons tlie question of accuracy is not 
of great iraportaiice inasniAch as each inde;: is checkeci, 
ajf^ainst th^^ ^.aie sdi .ol f^rades* ^.atoTer mireliability 
exists in %lq rmrks, therefore, iiill affect all corre- 
lations alike. 



TABLE 10.. GORREUTIOHS BETIEEI^I IIAEKS IN M&LISH MD THE 

IIIDEX YMIIABLES ON THE TIIIUr,E SCALES 



Scale 


Iferks in Enr^lish xvith 




Score 


Atteiorots 


Ei^ts 


Wrong 


Accuracy 


Otis 

Terraa:.! 
Ghica^^o 


+.56 

+.63 
+.52 


+.36 
+.3(^ 
+.B7 


+.57 
+.56 
+.53 


-.32 
-•36 


+.46 
+.46 
+.48 


Mean 


+,57 


+.31 

— -^"SS" 


+.55 

, 1 


-.33 


+.47 



TABLE 11.- CORRELATIOMS 3ETWSSN limG IN HISTOIIY MD THE 

mm. Y/J-ilABLES ON Tm THREE SCALSS 



Scale 




:-.ferk3 


in History with 




Score 


Attempts 


Rii^ts 


Wrong 


Accuracy 


Obi-s 

Terman 

Chica^^o 


+.53 
+.63 
+.48 


+.17 

+.24 
+.17 


II II 1 1 ■ 1 ri 

+.54 
+.6^ 
+.41 


-.45 
-.38 
-.37 


+.55 
+.53 
+.47 


H©an 


+ .D5 


+.19 


+.o2 


-.40 


+.5J2 



TABLE 12,- GOEHEUTIONS BETWEEN JilAEKS IN MA^ 
I>JDEX TMIABLES ON TI-IE TlffiH SCALES 



p. 35 

tlEIIATICS MD 



Seal© 




Marks 


in Mathematics mtli 


Score 


Atternpts 


Ri;^;its 


f'ran^ 


Accuracy 


Otis 

TeriTian 
Chicar^o 


+,60 
+.54 
+.47 


+,45 
+.32 
+,29 


+,61 

+.72 


~,17 


+«40 

+-36 
+.27 


Mean 


+.54 


+,-35 


+.53 


■*,ZZ +.34 

L ,n,. -l„i —- — » 



Correlations of the fi?e index variables i^ith Ehglish, 
History j,and Ifathetnatics ars r^-X^m in Tables ^,w,.^^,and IS 
respectively.. Inspection of the avsrago correlations for 
these tables shows a close ao^reerasjit for the three subjects 
studiedo The coefficeints for Siir__...-. :""'jestjHiE'tory 
next, and Lkthernatics lowest/Dut the diflerences are sli^^iti 
In all three tables Score has the highest correlation 
mth school imrks. The next higjiest corrslations in order 
are Eir4its,AcGijracy,WrongGjand Atternpts. This is precisely 
the order found in the section on reliability. Thus if 
school uork be measured by luarks ti^e s -„^t,tiTe merit of 
the T0.rious indexes for prediction is SjH.a W,and A. As 
in the precedin?^ sections, correlation invoiring S and B 
are aore nearly equal than the otiiers. 

Section 10. Suiimary for Reliabi lity of Index es for Ihol© 

Scales 

For \iiolG scales consisting of batteries of tests, the 
authors' formulae appear to be sli^tly superior to total 
Rights as as index. Table 13 gives the avera^^^e correlations 
and differences in f^vor of S (absolute values considered) 



p. 36 



TABLE 13.- SirtiABY OP G0RPJEU7I0NS FOR SCORE AMD RIG-IITS 



Yai-'iablcs Correlated 



AYers/'te 
Coefficient 



DiiferGnce in 
FaYor of S 



Score and Riplits +.^^6 

Scales Indexed by Score \ +.76 

Scales Indexed by RlQ:hts i +.74 

Terman Forais B and A 'by S ; -i-.i?! 

Termaji Perms B and A by R | +.;)0 

Ai^e ?7ith Score i -.-^U 

A^e i^th RiF^ts I -.38 

lErks ?.ith Score j ■^.•'^5 

Marks mtli Kip-hts 1 +.b3 



Total Difference 



.02 
.01 
«03 



.08 



The extreme sinr:"icity of scoring?, by Pj.r;:hts,howeTer, would 
seem to more than outweir';h the slic'^t advantage in favor 

of niorG Go^ir^icated formalae* 

Accuracy has been shorn to have the pecaliar e^vant- 
af^e of being unaffecte<^. hy speed, and at tha seae time to 
posser- -■-•-. •"'-'^^'-■'. . .,v, "■■'') Guismarsj: cc.-w, tions in 
favor of Score are shovin In Table 14. The total differ- 

TABLS lA," m^^^MCf OF nomyHLATIGNS FOR SCQIS AliC ACGURACI 



Yario.bles Correlated 


Averr!.'-''6 
Coefficient 


Difference in 
Favor of, S 


Score and AccLtracy 


+.76 




Scales indexed by Score 


+.76 




Scales Indexes by Accuracy 


. 70 


» . « . 03 


Terrnan Fomis B raid A by S 


"1 » t/ *^ 




Terman Forms B and A by R 


+.84 . . 


- , . . .07 


Age ^dth Score I 






Afse iPTith Accuracy 




1 "^ 

1 A . • » X :' .< 


Marks i,ith Score 


+.55 




Marks mth Accur-acy 


+ .44 . . 


f . . . .11 


Total Diff'^-r 


ence 


. 00 



P* 37 
^ces in favor of S are ohowk in Table 14, Tlie total 
differsncea in favor of S indicates that Accuracy is som - 
T^at less satisfactory than R accordinp^ to the criteria 
eniploy-30.. Moreover it is a more linvolved complicated index 
than R, but not so involved as S# 

The results for Errors are -nresented in Table 15. Th® 
general merit of Wrong as an index is less than that of 
the preceding variables. Erroifs, however j have a surpris- 
ingly hig^ reliability and are utilized to advantaf;?;e in 
formulae discussed in the following sections. 



TABLE 15.- SIPriAJiY OF CORPIUTIOHS FOR SCORE MD ERRORS 



Tariables Correlated 



Average 
Coefficient 



Score and WroA<^ 
Scales Indexed by Score 
Scales Indexed by Wrongs 
Terman Forins B and A 1^ S 
Terrnan Forras B and A by W 
Age With Score 
j^e With Irongs 
mrks With Score 
Marks With Wrongs 



4 o « c 9 o 



-.58 

•¥,16 

+.69 
+.91 
+.74 . 
-.41 

+ . *j^- » « • » » a i 

+.55 . 



o » o » o 



Total Difference 



Differences in 
Favor of Score 



.17 



'i o a » ^ 



.68 



Attempts , ?Mch are frequently used as an index for 
tests 5 appear to have the least merit of any of the var- 
iables discussed. Table 16 gives the averages and differ- 
ences as in the above tables. The total absolute differ- 
ence in favor of S is greater than for any of the pre- 
ceding variaiiles. 



p. 38 
TABLE 16, - SUWMEY OF GOEPvEUTIONS FOR SCORE MID ATTEMPTS 



Yariables Correlated 



Score and Attempts 
Scales Indexed by Score 
Scales Indexed by Attemnts 
Tei^nan Fornis B and A by S 
Termaji Forms B and A by A 
Ag© with Score 
Age with Attempts 
mrks ?ath Score 
Marks with Atteiirpts 



Coefficient 



+.76 
+•56 
+.9i 
♦.68 
-•41 
-.26 
+.55 



• « 



Total Differsnc© 



Difference in 
Favor of Score 



» « p 







« a tt o 



o d G « 



.20 

.27 



Bi-rSi3 Si 3 cri?ninati T8 CaT-.-- 



u:.2.0 Ind©2C93 



In addition to the general, reliability pf an index, 
another valuable propeir!" ' r" sudi " — 'i.able is the extent 
to which it makes possible discriniina,tion between indivi- 
duals ezid between groups rixon real differences exist. A 
test Miich reveals too narro?/ a range for a given group 
fails to discriminate between the individuals of that group. 
Such undistributed score is a defect in the test or in the' 
mode of indexing. Similarly a test or raode of indexing which 
fails to discri.-.iinate between groups is defective if the 
characteristic is in reality different in tjrpe from group 
to group, ©lus a test #iich shows all individuals in G-rade 
5 to possesB the same ability, and at the same time reveals 



p. 39 
no difference between mean scores for Grade 5 and 6 is 
lackin,<5 in individual and in group diseriininationf Th© 
fundamental assuniption is, of course, tbi?.t such indlYi- 
duals and gro-"nr. do Tary and that failure to detect the 
variations lies in the particular mode of indezinr?- tiie 
trait in question. 

Section IX Carac ity of the Indexes to Pi scriroinate between 

IMiYiduals 

Disoriraination between individuals of a group is best 
studied by meanss of frequency distributions. In the pres- 
ent study J however J such an elaborate method as this is 
unnecessary inasmuch as intelligence tests are ©f suffici- 
ent length to give a fairly good spread for all indexes, 
!aie distributions for S,1»A»W, and E» in the Appendix are 
typical of those for all tliree intelligence scales, fhe 
standard deviation for these variables are given in Table 17« 

TABLE 17.- STAJ^ARD DETIATIONS FOR TERIIAII f'OMS k Mm B 

mm aRoup i high c 



Tariabl© C^i (first) 



(o;^( second) 



c^~^ iDiff^P.S.di^ 



Score 

Attempts 

Ri£^lt3 

Wrongs 
Accuracy 



33. 79+1 ,39 
24a 250. 99 
26, 9551 til 
19,73+0.81 
f. 13+0,01 



30.91+1,27 
20*63+C,85 

23.5^r^,97 
19, 66+?^, 81 

0.127^.01 



2,88+1.68 
3,49+1.30 
3.41+1,47 

0.07+1.15 

o.oiTo.oi 



1.5 
2,7 
2,3 

1.0 



1!here is sowe evidence that there is less variability in 
perfomaance on the second trial (Form A) than on the first 
( Form B), Tliis is incidentally a bit of evidence to the 



effect that equal practice for a ^oup of pupils tends to 
bring them raore closely together about a central t.^ne, a 
result contrary to that held by some psychologists. The 
differences, however, are slight, altho^ji/y^i in one direction, 
and the tw) fomis of the test i!i8.y not be equivalent for this 
purpose. The result is then merely sugp^estive. 

^e standard deviations for A,R, and W in Table 17 ad- 
mit of direct comparison inasziach as they are all ezxDressed 
in point or response units. The order of discriminative ca« 
pacity for these variables is then R,A, and W. The indexes 
S Kid. I .-^re expressed in different units and hence may not 
be compared mth the rest. Considered on the point basis 
however 5 author's score has the greatest capacity for dis- 
crimination between individuals on account of the 'jveigiit* 
ing and formulae invslved. The standard deviations -for Grade 
7 and I High B are also given in Table 18. Tbe results ag- 
ree with those of the preceding table. 



TABLE iw.- STMDAI® DWIATIQNS FOE THT; TIimHE SCALES 



Scale and 


Stsndazxl Deviati :.i] 


L.g for 


"■"" - -"-'■ ■"'■'■'-- - 


Group 


Score 


Atternpts 


Ri^^its 


i'rong 


Accuracy 


Otis G-rade 7 


19.91 


18.00 


19.92 


13*46 


0,06 


Otis I I-Iigh B. 


22. 4n 


18.02 


21,55 


17.02 


0,^8 . 


Termeji Grade 7 


23.52 


19.17 


16.52 


12,46 


0,08 


Terman I Hif^- B 


26.87 


18.39 


22.34 


14.87 


0,09 


Ciiicago Grade 7 


i^.sa 


6.20 


6,16 


^.60 


0.09 


Chicar^o I Hif'-h B 


12.03 


5.8^ 


8,34 


6.50 


f^.l<5 



p. 41 

In order to study the variability of a group by a 
statistical measure independent of the units employed, 
Pearson's Coefficient of Tariation, V - '^^^'^ , 

was ©nployed. ISie results for two groups appear in fable 
19. It is at once apparent that #iile T is independent 
of the units employed it may nevertheless le$d to results 
which exQ confusinin;. The largest coofficients of variation 
are for f , an index ifnich migtit readily be supposed to 
furnish the least variability. The result is brou^t about 
by the relatively large standard deviation of 1 (Table 18) 
and. the relatively low mean (Table ,1!f>, below). 



TABLE 19.- COEFFIGIMTS OF YARIATION FOE TI-E THREE SCALES 



Scale and 


Coefficients of fariation for 


aroup 


Score 


Attempts 


Rl^ts 


Wrong 


Accuracy 


Otis arade 7 
Otis I Einh B 
Texiaan Grade 7 
Terman I Hi,^ B 
Chicago G^rade 7 
Chicago I Hig^i B 


14.2 
14.8 
18.8 
18.7 
19.5 
21.7 


1C.2 
9.5 
13.8 
11.5 
11.7 
10,1 


14.3 
14,3 
16.6 
17,4 
14.6 
18.8 


36.2 

45.6 
45.5 
47.7 
43.0 

48.1 


9.86 
1^.35 

9.85 
11.37 
10,71 
13.18 


Iloan 


17.8 


11.1 


16,0 


44.3 


10.89 



The coefficient of variationj depending as it does upon 
the position of the distribution on the scale, is likely 
to give a very rnisleaxiing result for distributions such 
as these above , and should in general bo avoided for com- 
parison^ of tliis type. 



p. 42 



Section 1"^ CaTiaQity of t-ie Indexes to Discriminate 

Detweon G-rou-',')S 



Table 180 gives the means on the tliree scales for 
Grrade 7 and for I Iligli B. It is at once eTident that the 
second group has the hidier mean for nearly all of the in- 
dexes. Accuracy J howeTorj appears to be nearly conaistent 
for all three scales and for both ^'oups. From the stand- 
point of discrimination, therefore, this index is of littlo 
Talue. The correlation ta,ble3 in tho Appendix show a con- 
siderable spread for Accuracy liiile the constants froin Ta- 
ble 17,18, and 19 indicate the extent of fiis variability. 

TABLE 20.. !IEMS FOP. THE TKREE B'lTELLIG-MCE SCALES 



Scale and 


ilGon?^ for 


Group 


Score 


Attempts Ei:9;rits 


"ITrong 


Accuracy 


Otis arado 7 
Otis I High B 
Termezi Grade 7 
Terman I Hidi B 
Chicago Grade 7 
Chicago I Higli B 


139.8 
151.3 

125.0 

143.7 
54.0 

55.4 


177.*" 
189.3 
138.8 
159.5 
52.8 
57.2 


139.6 
152.0 

111.4 

128.3 

42.1 

43.7 


37.2 
37.3 

37. 4 
31.2 
10.7 

13.5 


0.80 

0.80 
0.80 

0.81 
0.80 
0.76 



Individimls within a group, then, differ considerably in 
accuracy. When the above inter-group comparison is made 
however, Accuracy is found to be relatively constant. 
These results indicate tliat the groiiili curve for accuracy 
ordered relative to a^e ¥dll be relatively flat in com- 
parison with ordinary score. This iriiole matter ?/ill be 



p. 43 

fully treated by the ^^.Titer in a forthcoming article on 
C3-rowth Curres under Different Modes of Indexing. 

In order to bring out such inter-group differences 
more clearly they are presented in full in Table 21. The 
quantity p.ir.a>>|-. denotes the inter-raean difference divided 
by the probable error of tliis difference ealeulated in the 
manner explained in preceding sections. Such a quotient 
gives a conTenient indez of discrimination. Indexes less 
than 2 or 3 sho?j that the discriminatiTe capacity of the 
test for such variables is not sigrdfican.t. 



T£BLE 21.- DISCRIMIIATIYE CAPACITY OF THE INDEXES AS SHOWN 
BY INTSH-r;ISAI\I DIFFERENCES IN GRADE 7 AND I HIGH B 



Scale 



Otis 

Terms2i 

Chican-o 



Inter-Mean Difference and Probable Errors 
for 



Score 



Ul 1 1 



.P,E, 



IS. 7 



2,7 
3.2 



Attenr^ts 



Diff.P.E 



12.3 
4.4 



3.4 
0.8 



Diff ,P. 



ia.4 

16.9 
1.6 



2.7 

a. 6 

0.9 



Vi/irong 



Diff 



.P.i; 



0.1 
3.8 
2.8 



.n 



1.8 
0.7 



Ac cur a c:^ 



Diff. 



0,0 
^.01 

-0.C4 



0.01 
0.014 



Ave. 



D 



3.7 






4.3 






t . ij 



Ttie five vsxiables in ordGr of their capacity are A,E,SsWj 

and R. Thus the groups studied show the greatest difference 

f 
with respect to speed sjid the least with respect to accura- 
cy. This result is quite in agreement with coiamon teaching 
experience. Pupils can be easily made to hurry, but it is 
exceedingly difficult to train them to be accurate. 



p. 44 
Miilo A shows the best ca,pacity for inter-group 
discrimination, it is not superior to tho other vaxi- 
ables for differentiatiA^ indiYiduals. Score and Rigiits 
again appear to be superior to tho other indexes for 
individual discriioination, a property rfhidi is raore inrpor- 
tant than inter-group differentiation. 

Section 13 Practice Effect with Repetition 

Form B of the Termon Scale was given to .'rroup I 

Hi^i C and Form A of bhe ;iajis test given ui^j r oilowing 

day, Assuining that these tT70 forms are equally difficult 

a practice effect for each of the variables may be noted 

as in 1!able 22 • There is a positive difference between 

the means for each of the variables except f. !Eliis last 

nex^atlve difference also means an iraprovement on second 

trial, so that the practice effect is imieated on all 

of the variables. The last coluim shov/s the sif^ificance 

of this gain. Tlie indezes R,A.,S5 and R, reveal gains 

A 
tliat almost certainly cannot be accounted for hy chance 

fluctuations, rdiile tlie change in W is in hanriony with 

that of tho other variables. Errors orA Accuracy show 

changes of less significajice for practice effect. 



m 



TABLE 



EArlS FOR TSEiIAI^ F0KI3 A AND 3 I'lTH OliOOP I HIGH C 



?ariabl9£ 



M A ( second) M e, ( first) 



Ma ~ Me | Dlff-vP.E.db.^ 



Score 
Attempts 
Hint's 
Iron?:; 

Accuracy 



101,74+1.79 

1^2.41+1.37 
58.59+1.14 

^■.63+<^.^l 



84.11+1.96 

146.19+1.4r^ 

87.15+1.56 

61.f^4^1.14 

0,59+^.^1 



17.63+2.35 
1.3.61+1.84 
15.26+2.08 
-.S. '45+1. 60 

o.^'4+n,r>i 



6.7 

7*0 

7,3 

V'l.S 

4.4 



t* 45 

Part II 
jjgalysis of tho Index Variables by ComDahgnt Tests 

^e authors* plans of scoring giYen in Table 3 show 
that 9 of the tests niakinr- uv. the Otis Scale are scored 
by the f orimila S R. Thsse nine tests ?/erG therefore cho- 
sen for analytical stiidy. The 3tabilit3" of correlations 
for whole scales has been shomi in Part I. In the folloT/- 
ing sections the inter correlations of the coiirponent tests 
show a M:^].i degree of consistency. The coefficients in 
general are loF/er than for wliole scales but they indicate 
the sarsie relationships between index Tariables. It will 
also be shofai that pooling tests increases both tlie val- 
idity and the reliability of the indexes j an effect which 
may be roughly forecast by certain predictive fomialae. 

Section 14 I ntercorrelations o f Variables for the Otis 

Oonroonents 

The correlations between Index Variables for the 
saiae corrqionents axe given in Table 23. In the last line 
of the table the coefficients for ail nine tests pooled 
are given for comparison. 



p. 46 

TABLE 23.- CORRELATIONS BETl^M INDEX VARIABLES ON NTNE 
OF TIE CO:iPO!JSITS OF THE OTIS SCAL3 



Test 


G-rade 7 


I Hi,^h A 




AH 


A>W 


R^W 


A-^E 


A>f 


E-.? 


1 

2 
4 
5 
6 
7 
8 

9 
10 


+.37 
+.32 
+.72 
+.72 

+.58 
+ .55 
+.28 

+.5G 


+.:m 

+.43 
+.30 
+.04 
+.48 

+.5" 

+.33 


-.81 

-.IG 
-.42 
-.66 
-.44 
-.20 
-.66 
-.27 


+.19 

+.81 
+.62 
+.70 
+.47 
+.32 
+.56 
+.81 
+.26 


+ .4;^ 
+.55 
+.05 
+.12 
+.48 
+.65 
+.16 
+.18 
+.29 


-.77 

-.05 
-.75 
-.63 
-.55 
-.52 
-.86 
-.43 
-.85 


Mean 


+.61 


+.56 


— . -j.C 


+ .5"^ 


+.u5 


-.CO 


All 


+.75 


+.21 


-.51 


+.5.5 


+.21 


-.70 



It is GiridGnt that these are hi^^^ier than the meems of the 
nine correlations on conrponent tests except for Atterapts 
with Errors, in 7.hich case the pool H.Te-j the lower value* 
In certain cases, therefore, pooling or len^tliening the 
tests has the effect of increasinr^ the correlation be* 
tween the indexes. Tlie exception in this instaiice is worttiy 
of note as a warning against applyi-ng general rules for the 
correlation on lerif^iened tests. The hir^i de.c^ree of con- 
sistency in +;.- coefficients indicates tliat pooling of such 
components is a justifiable procedure inasmuch as the test 
material is fairly homogeneous for mirnoses of indexing. 

Inter correlations between Rlr^its on the nine conxoon- 
©nt parts of the otis scale are given in Tables 24 and 25 
for Grade 7 atid I Hi,^i A respectively. Both groups consis- 
ted of 50 pupils. All coefficients larger than three tiraes 



A 



p. 47 



their nrobablo errors are printed in heavy type. 



TABLE 24. ~ GORKSUTIQ-LS BS^HEEl^-I RIGHTS ON TIIE HII^ OTIS 

CO:!P0!-n5ITS for GPJU)E 7 



Tepyt 


1 


9 


4 


5 


6 


7 


G 


J 


iO 


X 




+.54 


+.40 


+.47 


+.50 


+.34 


+.33 


+.50 


+.34 


2 


+.54 




+.^Y 


+.53 


+.39 


+.49 


+.26 


+.27 


+.27 


4 


-^.4^ 


+.S7 




+,35 


+.43 


+.37 


+.4C 


+.38 


+.19 


5 


+•47 


+.53 


+.35 




+.51 


+.3C' 


+.35 


+.14 


+.16 


6 


4.. 50 


+.39 


+.43 


+.51 




+•56 


+.3^ 


+.36 


+.15 


7 


+.34 


+.49 


+*.'^7 


+*30 


+.5G 




+.13 


+.52 


+.35 


8 


+.33 ■ 


+.r36 


+.40 


+.35 


+.3:: 


+.13 




+.19 


+.29 


9 


+.5f^ 


+.S7 


+.38 


+.14 


+.36 


+.5S 


+.19 




+.38 


10 


+.^4 


+.?.7 1 


+.19 


+.18 


+.15 


+.35 


+.29 


+.33 




Mean 


+.41 


+ »--)0 


+.M 


+,35 


+ .4'^ 


+.'.y/ 




+.34 


+.26 



A siiUTile calculation will show that this includes all co- 
efficients numerically greater than .^7. In Table 24 only 
8 of the 36 correlations are not sif^^iificant, Miile in 
Table 25 the sswne mmber occur. Most of tliese low eoeffi- 
cients @xo foimd in the correlations Mth test 10, the 
mean value for whidi is lower than for any other test. 
This one conr)onent then appears to be out of hsxinony with 
the rest; i.e. to fail to measure tlie same thing as the otl> 
er teats of the battery. Inspection of the Otis Scale shows 
that test 10 is for memory^ a trait quite different from 
those involved in the other components. Except for this one 
test a fair degree of consistency is found for the coeffi- 
cients in both tables. The means for all 36 coefficients in 
each table are +.33 and +.36 respectively. 



■p. 48 



TABT.K 


^5»- COBREUTIOIIS Bm^'mm RIGHTS d^ THE mm OTIS 




C0ilP(3iWS FOR I HIO-I A 


Tost 


1 


1 1 
*> z. p» 


6 


7 


8 


^ 


iC 


X 




+.36 


+.23 


+.4"> 


+.5.^ 


+.38 


+.5'' 


+.42 


+.3^1 


2 


•f.ac 




+.41 


+.36 


+.42 


+.a5 


+.56 


+.47 


+.21 • 


4 


1 'SO 


+•41 




+.S3 


+.ri4 


+.36 


<¥^34 


+.38 


+ » vJii< 


5 


^'•45 


+•36 


+.23 




+*52 


+.38 


+.41- 


+.17 


+.33 


6 


+•59 


+.49 


+.24 


+.52 




+.4G 


+.5': 


+.45 


+.17 


7 


+.3GI+.33 


+.38 


+•38 


+.4^ 




+.4i 


+.:i6 


+.02 


8 


+.3v +,5e 


+.34 


+.44 


•¥•0 J 


+.44 




+.3i^ 


+a^ 


9 


♦.4'! 


+.47 


+.36 


+.17 


+.4D 


+.36 


+.3^ 




+.13 


10 


+,34 


+.r}i 




+.33 


+.iV 


+.^2 


+.i»J 


+.13 




Mean 


+.41 


+.4^ 


+.33 


+.37 


+ .'-a4> 


+.35 


+.4^^ 


+.35 


+.21 



Tables 26 axid "7 iIidw the correlations betueen errors for 
the GORFionent tossts. Only 14 of the 36 coefficients in 
Table 26 are sif^iificant, U%<3 mm for tlio nliol© table 
bein?? +.S4. T©ot 1^ shows uBxt to Uie lor:€3t averacje cor- 
relation TTlth tlie other teste, so tlio.t it is of little 
significance indozed by the ri^rhts or m-on^s. In Table 27 
the coefficients are soaortiat hifliorj the iiiean of the 
iHhole tabl© boinc^ ^•S^* Eiriit of tbe 36 coiTelations are 
si^ificantj rdth fi¥e of the lowest values a-^^^^earing 
Tdth Test 1". It is difficult to explain the difference 
in correlation for the two P?"aup3 Mien indexed by W« Ta- 
bles rJ4 and 25 showed moan values nearly identical , but 
the difference bet?/een the riean coefficients for W is too 
larije to be ascribM to chance. One explanation of tliis 
difference may be found in tlie fact that arour> I Hir^ A 
made more errors than Grade 7 ( See Table ) * The effect 
of this was to f!;i70 less jaminr'^ in the contingency ta* 
bles with a resultant hif^ier correlation. 



p. 49 



TABLE 26.- CORRELATIOMS BST^;.?EEN SFIROES ON THE NINE OTIS 









aO:;ffO 


!E^ITS 


FOR ; 


]iUDE 


7 






Test 


' i 2 4 


5 


6 j 7 


8 


9 


10 


1 


•^.16 +.171 1-.^ 


+ .11; +.29 +.3^"^ 


-.^1 


+.15 


2 


+.16 +*24 


+.15 


+.18 +.35 +.16 


+.17 


+.13 


4 


+.17 +.24 




+.3?. +.30|+.44- 


+.37i+.33 


+.28 


5 


1 +.49 +.15 


+.32 


+.18^ +.46 


+.35 +»09 


+.19 


6 


1 +.11 +.18 


+.30! +.10' +.40 


+.24 +.18 


+.22 


7 


i +.29 +.35 


+.44 


+.45. +.4^ 


+.43 +.21 


+.23 


8 


i +.3O1 +.16 


+.37 


+.35 +.24: +.43 


+.21 


+.14 


9 


^ ".n^ +.17 


+.33 


+,09 


+.18i+.21 


+.21 


+.12 


10 


+.15; +.i;-i 




+.19 +.^': 


+.^3 


+.14! +.12 




Mean 


+.'51 


+.19 


+.31 


+.^^8 +.23 


+•35 


+.27 


+.16 


+.18 



TABLE 



CORREUTIOnS 

COIPONEfWS 



ITEEiy EimOES on THE NINE OTIS 
FOR I Hian A 



Test 


1 ' 2 

* 1 


4 5 


6 


7 


8 


9 


10 


1 




+.44 


+.42 +.28 


+.36 


+.58 


+.4i/ 


+.46 


+.39 


•J 


+.44 




+.44 +.36 


+.54 


+.57 




+.51 


+.18 


4 


+.42 


+•44 


+.3^ 


+.37 


+.54 


+.5i 


+.39 


+.22 


5 


+.2G 


+.36 


+.301 


+*4e 


+.52 


+.46 


+ .12 


+.29 


6 


+.38 




+.37 : +.46 




+ .-4*0 


+.49 


+.26 


+.16 


7 


+ . .JO 


+.57 


+.r34!+.52 


+..13 




+.54 


+.50 


+.31 


8 


+.49 


+.36 


+.51 +.46 


+.49 


+.54 




+.21 


+.24 


9 


+.46 


+.51 


+.39 


+.12 


+.26 


+.5C 


+.11 




+.10 


10 


+.39 


+.18 


+.22 


+.29 


+.16 


+.31 


+. 24 


+.10 




Mean 


4.4:j 


+.42 


+.4^ 


+.35 


+.4^^ 


+.51 


+.41 


+ .32 


+.24 



Conroarison of the four tables abore slio^s that the 
inter correlation of errors on the 9 component tests is 
about as hi.idi as for Blfrhtz. It may be noted also that 
none of the correlations are as hir^h as .6 uhile the means 
in all the tables are less than .4. Such coefficients are 
not considered hif^. The correlations corresponding for 
■Biiole scales as f^iven in Table 6 are +.74 for Ri,'^its and 
+.69 for Errors. Clearly then the cwjalating of tests to 



p.50 
form ?jhat haa hem called a scale score has the effect 
of raismo; the correlation or^ in other words, lerif^thening 
a test increases its reliability for these indexes. A more 
detailed discussion of this point rail ari-ear later. 

Correlations bet?/8en atterrr^ts on the nine corar)onents 
have been worked out for one group and are given in Table 
28. Of the 36 coefficients, 27 are significant, the low- 
est averaf^e again occurring for Test i«^ with each of the 
others. By tliree modes of indexing, then, -this test shows 
up as distinct in type from the rest. The ;Tiean correlation 
for the whole table is -^.32 which may be coiTipared with the 
mean coefficient of ,56 in Table 6. Lengthening the test 
also increases correlation ',ihen Atteirrrts g^o employed as 
the index vazlable. The coinparison iis only a roug^ one, 
hoT^^ever,; for soriiewhat different scales and groups are em- 
ployed in the two cases. 



TABW. ^!8.- UORRELATIONS BST^EM ATTEMPTd 0>I THE Nimo OTIS 

COAlPOim^ITS FOR I : HlOIi , A 



Test 


1 2 ■ 4 


5 


6 1 7 

1 


8 


9 


ir 


1 




+.41 +.^-e;+.r;8 


+.35 


+.4^ 


+.21 


+.31 


2 


■j-.S^' 


+.3^1 +.13 +.38 


+.41 


+.3y 


+.:^9 


+.S6 


4 


+.4li+.3'l 


+.41 +.S9 


-^-.29 


+.32 


+.35 


+.20 


5 


+.48 


+.13 


+.41 




+.31 


-i-.^g 


+.34 


+.37 


+.18 


6 
7 


-{-.ac 


+.3a 


+.?.9 


+.31 




+.52 


+.27 


+.4<^ +.26 


+•35 


+.41 




+.29 


+.52 




+.37 


+.22 


+.33 


8 


+.4C 


+.39 


+.32 


+.34 


+.27 


+.37 




+.40 


+.26 


9 


+.21 


+.29 


+.35 


+.37 


+.4^^ 


+.32 


+.4^ 




+.C8 


i^ 


+.3i 


+.26 


+.20 


+.18 +.26 


+.33 


+.25 


+.C8i 


Mean 


+.34 


+.31 


+.32 


+.31 


+.34 +.35 


+.34 


+.29 +.24 



In Part I, Table 5, B and W shoiTOd a moan correla- 
tion of -.49 for 'fstiole sco^les. The oorrQ3;'-ondii%^ coef- 
fioiento for the Otis eonr'^^^^n--^ are giTcn in Table 29 
with a nieaii of -•'^1. The Inoirease in correlation b^ pool- 
ing is a'?pin STidont. The oorrGlG,tions in tha principcil 
diagonal aro betweon Rights snd Errors on the sains test 
and are therefore lar.<^r than the rest, Uis mean bein^ 
-♦48, The re!TiainfIf?r of the table r^iverj correlations for 
all poseible co.nbinations of Hi^^its anci Wrongs on the nine 
tests two at a timo. All biit tliree of tlie 81 coefficients 
are ner^itiiro Miile just one third of tJiea axe si;<5.iif leant 
aoccrdin/^ to tiie usual rale. 



TABLE 2^.- UOBBEUTIOHS 



BIGHTS AHD SRBQHS ON THE 







DTI3 OG . . 




FQIi OIL^DS 


7 Alffi I HIGH A 




Ei'-'hts 






Irongr^ 


•* 

^ 


.■■, 


4 


;) 


6 


7 


8 


J 


10 


ilean 


1 


-•8Jl 


— *t5 


-.'M 


-.^4 


—33 


-.41 


-..T/ 


-.rJ8 


-.31 


-.35 






-.18 


-.17 




-.15 




-,r 


-.r 


-.25 


-.17 


4 




-.11 


-.•l*^ 


— . *- 


-.11 -.^-0 


1 r^ 


-.f^9 


-.OS 


—13 


5 


-.^M- 


•.27 


*.4^ 


-,66 


-•44 


".29 


-•Dl 


-.^7 


-.la 


• .36 


6 


-.U 


-.^a 




-.^9 


-.4'4 


-.'^■^ 




-.S^ 


-.11 


-.Id 


7 


-.36 


-'.i^ 


(1 1 


'■■J 'T 




r. . 






8 




-.22 


-.42 






-.17 


-.Cc 


*.<^7 


-/^6 


-.28 


'J 


.,<^4 


-.C?^ 




^."'^ 




— • 


-.1-^ 


-.^7 


^.n 


-.^y 


i^ 


4..'^7 


+.^4 


-.^9 


-.^7 


-.If) 


-.17 


-.13 


-.08 


-.54 


-.13 


!iean 


-.•:;) 


' ft 


-•^1 


•»•■■- 


-.23 


-.19 


— . :.0 


-.14 


-a^ 


-.21 



ThiF5 au^T^Gnts tiir.t the criterion of ihroQ times the -irob* 
able error i"; too ntringsnt for tests of this kind. If 
tiTice th© rrobablo error viere edovted in Uw present case 
all ooofficients over.lO vjonld be sL^ificent including 



p. 52 

five ninths of the total nui'nber, while all of those great- 
er than one probable error or over .l*^ ?;ill include 56 out 
of 81. For the last case 25 coefficients are less than one 
probable error, yet 22 of them are negative in sign. Thus 
a coefficient less than one probable error appears to give 
assurance of negative correlation beyond the expectation 
from the usual rule of even chance; i.e. the probability 
of significance from the data appears to be greater than 
by theory. In any case, highly consistent negative corre- 
lation is exhibited by the whole array. 

Section 15 Correlations of the Otis Components with Age 

The correlations of the age factor with each of the 
Otis corrrponents are similar to those for whole scales. Ta- 
ble 30 shows higlier correlations for the nine tests pooled 
than for the mean of the tests. Here again the effect of 
adding tests is to increase correlation. The formula for 
estimating the correlation of the pool of "n" tests with a 
criterion may be written in the form: 




Considering l,u^~ - r,o as Rights for Otis G-rade 7 and age 
as a criterion, the constants in Tables 24 and 30 give for 
this coefficient, p 

a value identical with that obtained by pooling the nine 

components. 



p. 63 



TiJLE 



3^,- COERSLATIOHS BmiEm ASE AM) THE imU YAHI- 

ABLSS ON TilS 0?I3 SCAIJi3 FQH QBME 7 Ai^D I IlIG J A 



.itt"!!..'!:.',, ;i! 




Grade 7 


«_ TE » Jl, u,^...^ 


I liir 


ii A 




Teat 














Ajt50^A 


Af;e*R 


AgGM',' 


An;G>A 


Ap:BkR 


An;e.<i. 


I 


*" « - •»-> ' 


••» Ji> 


•?-,lw 






4., ^2 




^,29 


-•»43 


+»I6 


-r.} 


•.•3i 


4", 26 


4 


-3" 


-•24 


-r^7 


-• 3. J 


-..29 


+•21 


5 


■"•♦CfU 


-•42 


"h.S3 


-•3;^ 


-47 


+.23 


6 


—23 




•^•'^5 


-•f^l 


-.6?! 


+.33 


7 


-..^u 


-^G 


4»^i 


«•• ■ .;, 


~.S6 


+.19 


8 


-.^« 


—'^6 


**^6 


••aii^ 


-.*4i 


+.33 


9 


i ■■-; 


-.::.: 


^.«^ 


••S3 


-.-^14 


+•24 


1^ 




-I-.C3 


"..35 


—« •-»',; 


■f*''"l 


•.04 


Lfean 




-•?6 


^•'^a 


•••f2v> 


-.3-^. 


+.2^ 


All 




«»♦ 3'.^ 


+•07 


-..j2 




"f.34 



Siriiilju" coefficients are piroxk in Table 31 • The differences 
between -predicted raid actual Talues are lii no case sif^iifi* 
cant* Tlie above forrmla, then, appears to be & useful on© 
in preuiotiiif, \ho Yaiidity (correlation mUi a criterion) 

of tests by poollrig oouiponeiits* ^^ :l-: U uo noted also that 
the formula will hsT© hi#i values for large values of J^* 
axid snF.ll values of Ax> • 



-y.e- 



mSLS i^l.- PBEDICTED AH) ACTUAL CORKEIATIOIIS BETWEEN AGE 

A!'^ I!'1DM YARIABIJ] 



(Jroup 


Variables 


Predicted 
Yalue 


Actual 

Value 


Grade 7 
I Mr^a A 
Crada .7 
I Mmi k 
1 Higi A 


RiHits AfT.o 
Hl^iits Ar^o 
Iro'nrs A^ 
Wropif-n Age 
AttQ';?p^ts A-^e 


-•39 

Ai "s 

+,--'i 


+.34 



p. 64 

To obtain a scale of hir^ Tf.lidity, therefore, ca^^onent 
tests should be f^elected T^hich haTs hif^i correlation id-th 
the oritarion but loi? corrslation a^nenrr thorasolves, Thom- 
dike Justified tlie use of tests Tdth 1o?j inter- correla- 
tions on tlie ,<TO-and tlmt they arc repetitive; i.e* meas- 
ores of tlie saiie fact* Sio aboTo fomiilaj honrevor, will 
give higli validity because tlio iiitor-toct oorrolations oc- 
our in the deiioninator, ajxl low vrluoG i.lli thus raise tho 
TaluG of thG fraction* The baois of toot soloction, theji, 
a-ppearo to be nmthoaiti cal , rr^thcr thcui psydiological* For- 
tunately 9 horrcTCr, the ti^'O baseo s^^t^sc. 

Se-ction :IC ^xo AT>-nli--.?:tiDn of Hel iabilit:: roiTiulae to 

In the proeedinr^ section it Tms sho'sm iheX pooling 
coniponent tectr. has the effect of increasing; the validity 
of the tots-l Goale; i.e. to the extent to viiich it corre- 
lates Tith a eiiterion, Tho TK)olinp; of tests ?Jill next be 
shoim to have a siimla^' effect tt?ion ih€ roliability of a 
scale; i.e. the coixelation bstT/een te/o for^'os of tos same 
tost. , 

It irlll bo iecallec. f*b,ai tv'o foiri^- of tlie Terwn 
Scale viQTo {rlTGn to Group I El^i G on successiiTo days. IU»- 
liability' coofficionts have been calculatcxl for ©ach of the 
l^ co'rrir^onsnt tests and fox* all oombinod. to r;;ivo the total 
score. The roBuJis sxe given in Table 32. 
lieniKjirs of the national Acsslei^y of Seisnces^ Yoi.X?? p. 316 



p» 55 

TABLE as*- RELIABILITY COSFPICIISITS ?0H TIIS TERLIAM SCALE 
BY C ; Aim TOTAL SGOBB 



f«Bt 


Porriala 


Correlation bet?feen 


Rank 


""""'"■' '" 






For!;is A and B 






1 


11 


+•630 


7 




z 


m 


^*Q^9 


3 




3 


II-W 


•f.G32 


6 




4 


B 


W-J^^ 


1 




5 


aa 


+*a52 


S 




6 


H-W 


+•482 


iO 




7 


R 


^••683 


3 




8 


R-f 


+.53^ 


8 




9 


H 


+.5X4 


9 




10 


:":R 


4-. 7^2 


4 




Memi 




4.. 679 






All 




^•9ir 







Certain of the indlTidual tents reveal a hi^p. decree of 

reliability» eopocially test 4 (lor;icrl selection) with a 

coefficient of .u'-^. It will also be observed *toat soae 

of the loT78st corroletions occur ?/itti tests scored R-?/» 

This point will be dealt TTith inoro folly in a follomrig 

section. ThQ moan of the reliability coofficionts on the 

tests is 'i-#67^, while ttie oorrole.tioa for total Score on 

the t~ra forms is -f.^lC, so timt pooiincj tho tests has tiie 

effect of increasim reliability* 

A •nrediotive foi'miila givon by Bro?aa, am also implied 

" b 
in Spear>;Kiri's G-eneral Tlieoreta may bo given in the foiia, 

irtierc n.. iz the correlation betwoon t7;o tsr;ts or the avor- 



a. fillia-.! Broimj, Essentials of Ilental 'leasurement, 6am- 
bridn-e diversity Preso, London, 1911 

b. C. Spearman, Correlation of Sims m&. Bifforancon, Brit, 
Jour, of Psy*j ?ol-5, ,419«-^i36 



p. 66 

age of several J and N tho niinber of tests thus amalgama- 
ted. Ill tii8 present exa^imle the average correlation frorn 
the firat three tests is, IJ,, In order to predict the re- 
liability coefficient for 10 such tests, it is only neces- 
sary to substitute these values in the above formula giv- 

The value from actual amalganiation is .92. Simlarlyj a 
calculation based upon the average of all i" tests also 
gives 

The use of the forniula in these cases, tJaen, gives con- 
siderable over prediction. 

In order to test the applicability of Broivn's For- 
mula more fully and to analyze more fjilly the effect of 
pooling tests on reliability^ a inore detailed procedure 
is next eraployed. Reliability coefficients on emulated 
tests -are obtained in t?;o ways: Ta.Q scores for tests 1 
and 2 on each forra of the Ternan Scale are ad.ded, and. the 
correlation determined; next tests 1,2, and 3 are pooled 
and the t'TO forms correlated, and so on until all 1^ 
tests have been cuimlated in tiiis fashion. The ^^econd pro- 
cedure is to befjin mth tests 1^ and 9 and araalgamate in 
the reverse d-iroction. Tliese em>3irical results are then 
coiirpared mth theoretical values obtained by substitut- 
in,r^ r ^.68 and N from 1 to 1^ in Brorai's Formula. Table 35 
gives the results of this lengthy calculation. 



p. 57 

TABLE ..^.-.TimOKETICAL A!1D ACTUAL HELIABILITY COEFFIGIMTS 
OBTAIIM) FRO;.! BROM'S PQK'UHJl Ml) BY SUCCESSIVE GHULATION 

OF THi;: Tm TEiriAIJ COlIPO^^EnTS 



Number of Teots 
Cianuilated 


Theorotical 
Value 


Order of CirJjilation 


I to i^ 


10 to i 


I 

Z 
3 
4 
5 
6 
7 
8 
9 


+,68 
+#62 
+•37 

+•9^ 
+•93 
+.i)4 
+.94 

+»*?5 
+•96 


+•^4 
+•81 
+.87 

+.i?'^ 
+.80 

+.03 

+»i3y 

+.9X 
+.9^ 


+.70 
+♦79 
+.83 
+•86 
+.64 
+.86 
+.07 
+•07 

+«r 
+.9a 



Inspection of the table shows a roiii^i agroemmt in 
the tliroe series, Pirpro 5 iMch ii3 based on -tho table 
brings out tli© oor.imrxsons niore olssrly. Tlie thr©© curves 
show a I'apid initial ris© wp to 'i Ctifua-c-.tsd tests and then 
a moro .Tpracluai incrsaso to ^i© nnxintuLi valu©. The more ra^ 
pid rise, of the cuxie cumlated from testr. I to 1^ t^on 
the i-everso one, is.m doubt due to t^w ^'j-;-lei reliabil- 
ity of tlie first fer- tests as indicated in Table 33* 

Willie tliG ^onorr.l s.'^G^-ient in thb tuiree cunres is 



videiit 9 no verthel c , : :.: 



\..l'.,^^v.i. O i 



3 & very gIoot tcaridcmoy for 



the thoorstical cunre calculated for i ~ »C>Q to give an 
over T'.rediction beyond 4 or 5 cw^iilated tosts. This laay 
b® partly duo. to the unoqijai value>s of UiQ individual 
reliability coefficients given in Table 32, but itiatever 
the oaas©9 the use of Broim's Fortmila for prediction in 
a case of this kind is open to question* The equation 



re 



jo^ju^^^ ' ^ 5"7 







p. 58 
and corresponding theoretical curve indica-te that to get 
any desired degree of reliability with +1.00 as an upper 
llm3.t. it is only necessary to amalgaiiiate tests indefin- 
itely. Tliiro is, of course, absurd. The fortniia giTes an 
over prediction fairly esxly in the series of cuinulated 
tests. From the afooYe tables it appears that foui' or 
five bjrpical tests of the battery will ^ive almost as 
reliable an index as the pool of all ton components. This 
result would accoijnt in part for the hir;.i reliability of 
such tests as tiia Chicago Scale Gonsistin/;^ of only fiTe 
components. 

Txiis problem is one of great i.u-.ortoiioe in test con- 
struction. If intelligence can be indexed with almost as 
great accuracy by a short scale as by one twice as long, 
the savinp; in time alone is enonnous. Moreover if the 
short series can be shofm to be as valid as the longer 
one by correlation ?dth criteria^ the abbreviated method 
is fortiisr Justified. Inasnmch as no suitable criterion 
other than age was available for the present data, the 
check camiot bo rio^idly applied. Tlie p,r5 correlations by 
half and by 77holo scales, however, agree almost exactly 
(-.37, -.39), so that with age as a criterion, the five test 
"battery is as valuable as tb.e ten test scale. 



p. 59 
Section 1? Summary of Malysio of Condon ents 

The relationships found betiYeen index Yariables in 
Part I arc yerified for component tests. These coefficients 
are in general lower than by ?/hole tests ^ so that pooling 
has the effect of increasing the eorrslation between indexes. 
Inter-eorrelations betT^een components for R,f , and A reveal 
a hi^cph degree of consistency for such short tests but are 
less stable than for similar coefficients by whole tests. 
Furthermore the consistent batteries of correlations qybxi for 
R and 1 on different tests indicate a Mgii degree of homo* 
geneity in the tost material iltli the possible exception of 
Test 1^. 

In addition to raisinf;^ the correlation between index 
variables J pooling tests also has the general effect of in- 
creasing^ the validity and reliability of a scale within cer- 
tain lirdtB. Predictive formulae are useful in this connection 
but are likely to give an over-estiinate of the correlation to 
be expected by pooling. Moreover the physical endursnos of 
the children deterrriinep. the ma-ximora len^h of the tests at a 
sitting, s!o that the formulae are limited in application. 
The gain in validity and reliability is rapid on pooling the 
first few tests, but the point is soon reachefl i^tiere the ad« 
dition of similar material affects the correlations but sligiit- 
ly. The roBultn indicate that a battery of foiir or five care- 
fully selected conrjoonents tall give an inde:: mth substantial- 
ly the same reliability as a scale tmce that length. 



p. 6C 
Part III Scoring; Formulae 

Section IS The Linear Forra ^ S^a. ( _Jjt K' ) 
a. Formalae y/ith Kirdi est Validity 

In Parts I and II it has been Bhoim that the scoring 
fornmlae employed by the authors of the scales have little 
effect ir:ion the resultant scores -Khen a nuraber of coiiroo- 
nents are pooled. The Terman Scale T;ith the coinioonents scored 
by the tliree formulae , S-E, S^2R, aad 3-R-Wh« a corre- 
lation of ■^,J& with the score obtained by using 3-H on all 
ten cofiiponents (Tables 4 and 5). The amalgamated score then, 
is not Tery sensitive to such chan.'^Gs in the comrionent scor- 
ing f orralae and simple fonns are recorniiiended on these 
groujids. The 3in;<2le conrponent, however, is rduch more violent- 
ly affected by chan-es in the formulae enwloyed to index it . 
Chan,Q;es in iveigiits ?;hich affect the pooled score but slight- 
ly, id.ll be found to have a pronounced effect upon the in- 
dividttal cormonents. 

Table 4 which gives the various co.nnonent scoring for- 
roulae used by the authors of the scales , includes only for- 
mulae of the linear type; i.e. equations of the first de- 
gree in the variables employed. Tnese variables are K and 
W, so that the most general formula used may be T?ritten, 
(1) S ^ o-^R + ^w) = o^R.-t-c\N" 

■a^ere a, b, and c, are constants. It Tvas also noted in 
Section 3 that tlie relatiomship, A-^'^ + vT 



p. 61 
makes it ^-ossible to e3r-,ress this formula in terras of R 
and A or W and A. For.Tiala (1) , ho^^ever, has been so gen- 
erally erii^loyed that it r/ill bo ado'oted here for further 
analysis. Foriiiuilae expressed in terms of the other vari- 
ables may be obtained by substitution if they are required, 

The question irn^^iediately arises as to the best val- 
ues to assi,Qyi the constants a and c in equationl. A gen- 
eral solution of this problem raay be obtained by the rnetk- 
od of least squares. Values for R and ?; rro obtained for 
each of the N individuals of a given population. Assuming 
that a criterion, K, is the best measure of such determin- 
ation^, " set of M equations may be for.ned, 

K , r= CX, R,-+ C,\J\)| 
K N - uLn Rm + Cm \1\Jh 

T/here the K's, R's, and W's are knom, end a's and c's 
are to be determined so as to minimize tliG inconsistency 
in the equations T^T^aich is asswned to be due to iniperfect 
measui'ement* 

Nezt Vi.V^^ V,., -ill be la^ittsn for the differ- 
ences bet7?een K.^K. -- K ^ and the values obtained from 
the best determinations for the a's and c's; 

These differences or "residuals'^ are assumed, to be 



p. 6Z 
nonnally dir-tributsd. YMIg the assui-option is open to 
question for data of this type, it is nevertheless thw 
best that can bo made. The most ^.rote.blo Yalues for a 
and c next require that the sum 

Tlie re.iainder of the procedui'e consists in setting 

up the "nornal equations" in the usual rray. Trans f err in^^ 
the variables to their respective means, and setting up 
these Gquations givec, 

Since, ^n - — ^ , and ;c>^- 77^^^ ? these equations 

may be m"itten in the form, 

a- H^vij S;;: -*- C v^oo - A Kvxr <^ 

Solving these equations for a and c rj;XTQS, 



(31 o~ = .1? 



^, (UKvj[\a\Al -/vkyvj 






<-£P 



^-' (AVv^-0 



The value c rnay then be written 
a 

This last result has been obtained by Thurstone as a 

aT'L.L. Thurstono, A Scoring' ilethod for~liat?J Tests, 
Psy. Bull., vol. }:YI, 110.7, July, i0l9. 



p. 63 
value for C in the for.-.iil^ S^R+C^^' such that the correla- 
tion Aks io :■ jaajcinuii? i.e, C or c is determined, in 

a 
such a Tjay as to r^ossosn the hif^ie.^t rnlidity Tdth a cri- 
terion. The foriiEila for thifj coiroli-li-ju is, 

a 
Thurstone also niakes use of Yulo's ©quation f or l-u1» 

tinle correlation to obtain an ezpreosion for the hi^thsst 

correl^.tion rxth tlic linenr fon-rala S = B -^ Cf/* This re- 



sult nr.y bs -.^xittGn [ .R'^ha -^ Akvm - 2 A-kw Rica jlyvv^f 
(7) RkCuJxi-cw) =\J i —^ Ar\wr 

Thn -^.ctual procGciiiTG inYolvecI in detcrmininft the con- 
stants a end e rail then be as foiiov;ss 

X. GlTe the tost to a p^om md obtain alr.o the criterion 
of validity ri.rf8j.n3t which the formla is to be checkeri* 
2« Score the test for R anc' '^ •'^'^<-'' no"r-n.t~ the constants 
RkA, Akvm, n^v^, c^r^^ a>--i G^vw ; ^K la iioii required if 
For:uala (5) is arj:ployed). 

3. Substitute these l^.st resiiltr? in equations (3)^(4)5 
and (5) and obtain the forfiiolae Sc cx^l+c vjO ^ S^R+%:V\r 
( dif faring only by factor of pro-nortionr.lityj L). 

4. To riralict the hi^ihest eoiTrelc'.tlon obtriinable ivith 
these foroulac, substitute the conr'Uted conPtsjits in equa- 
tion sO- 



a. a,U.Yu}.G, Introduction to Statistics, C»G-riffin,. London 

1919, "48. 



p. 64 
b» Limitations in the Use of the Forriiala S -(x R-t-eVAT. 

In section 3, tv'o "?lans for ^'•''/^linisterinr; tests 
i^ere described. Accordin,f^ to the first, the ti:ae is fixed 
and Att0;:-!pts, Ri^^Jits, and Wronp,s allowed to vary; accord- 
inp;to the second, the mmber of Attev^^tc iz fixed, while 
Time, Fiip-hts, and Wongs axe recorded. T?:o of the index 
variables are thus alternately controlled by the method 
of adziiinisterin^, the test. 

The fonnula S- c^R-*- e\j\/ ^ ^--ith constants determined 
by hiHiest validity with the criterion, serves very well 
for tests given according to the plan of fixing the time. 
All of the tests eiTiployed in the tlii-ee intelligence scales 
are of this t^^nie and hencs no difficulty is enco^jntered. 

For tests adrainistered by fixing the Attempts, how- 
ever, the above fornola is inadequate. This arises from 
the fact th;;<.b S is no longer a firf-cti on of t?/o indepen- 
dent vaiuablGSj but of only one. If Atbeiiipts are constant 
and A-'^ir^ , then \/VJ - - R -t cH where d is ^constant. 
SubstitutiH;?: the value for \V in the eiqpression S^^P-+cW 
gives S^o^F-.t-c(d-f^') or S- Ca-^:)R -b cj^, Thus the 
seorin^^ fonuala is indeponcent of , and no fnatter what 
value is assigned to c (with the exception of c^o- , for 
which value tlie correlation is zero) the correlation /'-i^-s 

given by fornula (6) is equalfe/lkA. This last result fol- 
lows fro:n the fact that /Uco^^i^j- fU^ where a and b 
are constants (Theorem 1, Appendix) 



p. 65 
In other words if Attempts are constant, the soor- 
iiiirr foruiala S- ol R has the same validity with a criter- 
ion as aiiy linear fujiction of ^ and V\^ (?rith the ezeep- 
tion RrW for ^hieh Hks^o ), 

A further liinitation in the use of the formula S- Ri-CW 
lies in its sensitiTeness. The Yalue of G as determined 
by Fornula id) de-oends u-oon <s^^ and <^^ #iich in 
turn depend u:oon the proportion of Eights and Wcan^ in the 
group. Thus ?Mle the value for C is the best prediction 
for the particular group, the value ?.dll very likely dif- 
fer very rmterially in other nrrou-ns with diff erinp^ ■neTG&n.t" 
a,r^es of Wrongs. This means that the formla may be used 
with safety only (for tests administered with tiiae fixed) 
when the value for C has been detennined after the test 

has been p;iven to the Tiarticular r^rou-n. not before. Thur- 

a 
stone discusses the sensitiveness of the forsnula, but is 

unwillinn- to p^dmit its lirnitations. Table 34 which is based 

on extremely len;<jbhy and careful ealculabion, indicates 
mde variation in the deter/ninations of C for the 9 Otis 
component 3 on Grade ? and I High A. The average difference 
in the values of C for the two groups is .549 and in only 
one test is the difference less than 1. These results 
point clearly to the conclusion that the above formula b^s 
little ,3eneral merits i.e. the value for C from one group 
cannot be safely assumed to hold for another. 



L.L.Thur stone, loc cit. 



fmiM 34.- TALIBS FOR C IN KIE fmilU B R-J-C'- 
FOR G-HOiJl^S OiiMJH ? AIID I Hiaii A 0^1 '^^'^ HirC^CC 
THE OTIS SCALE (MS GIIITMION) 



F, 66 

:i!IED 
^3 OF 





GradG ? j I IIl'"^;.i A 


Tost 


,V-,,R 


R»\Ar 


i>-^)^v\r 


'-OR 


*0 \N 


'^- 1 


(^.R 


R^VvT 


(J^/VaT 


1 


«.38i 


-•8^3 


■^a?.6 


2.6^1 


?.,r^H 




-.r;7 


-.767 


+.^17 


Z 




•».176 


^•183 


S.S13 


1.794 


,1 " •• 


-.3<^6 


-.047 


+.m2 


A 

•X 


•♦•!..>■,.' i 




-•^66 


,1 1 0'-> 




•■'!'•• v/i/'r. 




-.752 


4-.S12 


5 


-.417 


-.66,^ 


^.2AB 


3.47V 




4^.16^ 


-.468 


-.627 


-►•:530 







'"•'^WC 


•?-.v-4.-ir 




2.^Y6 


•f^.o ^* 




-.052 


+.3'"^ 


7 


-•261 


«.r:76 


•^.214 


2»758 


S.C59 


-.6S4 


-.^■56 


-.516 


+.169 


G 


•* ♦ J "^ */ 


-•GVD 


-^.'^V^^ 


.'■i.a^^r 


r;,5j^7 


•f .4I-1> 


-.4^0 


-.857 


*.3r:7 


% 


^» Xv^.^> 


-.r;74 


+.rrc 


5.116 


.';.4r'7 


+.5V0 


— •'^i^iG 


-.4oy 


•f.S44 


X'> 


^.f^3C 


-.540 


-.M7 


s.9as 


g*5V9 


1.376^ 


>.0C6 


-.t^l'-.04^ 





!i^SLE 3^. 


. COI:TI 






Test 


I Kir-i A 








C5^ 


kf^vfj 


Cz. 


C, " Ct^ 




1 


i.^ie 


^.le^ 


-J-C.C5': 


^- ■:- p; 




2 


n.0^i4. 




-1.''69 


^.875 




4 


'^. 665 


3.665 




.r.C32 




5 


3.372 


Z.ATi 


*f^-.272 


-.112 




6 


%514 


^.^mc 


-^.l^^j 


^,^^'-^ 




7 


2,871 


3.57?? 


*^.267 


—347 




8 


3.83^ 


';.655 


^.im 


+.2:?5 




9 


4.593 


2.715 


-C'.P.VC- 


^#948 




1^ 


3.^:36. 


3.f>79 


^1.^36 


•i-1.14^ 





c. Tho Use of the For:isla S - (R t C ^ 



In tsst roaterial ^^mrQ t-vro altematiYos are given for 
each item aiid c^aessiiK tlierefore posaibl© the forrnola 

S- (R-W (i^.«X--~k) has been frequently adopted. Simil- 
ar f or:milae are used -mmi tli© nmmQT of choices is greater. 
These O2:prop.3iom5 are asBmiod to oorrect for guessing ele- 
ment Ira'-ol^od. According to tlio r.boro or;ur.tion a person 
f^essiiir; blir/dy on all of the itoms will get Ii^^-lf of thea 
ri^t by ch^^Gej antl hence & zero score v/hieh he deserves* 



imla penalizes justly; but it also penalizes for errors 
which are not due to guessiix^^, and hence unjustly. As ii 
result, for very difficult material, nearly all of the 
scores may be ne.niative. 

A noj'iif^oi^ of oxperimental atteiiiiybs haye been imde to 
deteriiiine the amount of fj^uessing in tests of this sort. 
After admni staring a set of Trae-False tests of the syl- 
logystic reasoning type , the v^iter asked the pupils on 
which items they had guessed. The number guessed ^vas about 
%-^jo per cent of tho total number of errors :aade, and of 
those items guessed only 8^ per cent T-ere LTong. For such 
groups and tests, the penalty attached to errors by the 
formula R-Wis enoniiously too rreat. It may be noted, how~ 
eyer, that the children did not always Iqiow whether or not 
they had j^uessed on any item. Reasonin^^ and guessing are 
often iiidistinguishable, and who has not credited himself 
mth reasonoiig when he has only made a lucky guess. 

Instead of assurninp* that a penalty should be attached 
for guessing, Thurstone proposes to use the forinula 
with the value of C to be deteiinined according to validity 
as above. This method at-ynears to be ^referable to that of 
a priori ^.a:ror.aixic.vio-i, wi-C'i os::- . _. . ministered wita 
the tine fixed. If Attoinpts are fixed » however, the foraa- 
la beco.aes independent of 1 as has just been shoivn^ and 
all values of ,C (ez-^cept G~+i) givea the same correlation 



p. 68 
This point is of great practical irrrr^ortance because 
most tests of the True-False type are administered so that 
all imy finish; i.e. Atterjpts constant. According to the 
above repjultsj, all fomiala,e of the type"^-^-:^ vv/ give the 
same correlation with a criterion as is obtained by using 
Eiglit'i alone? i.e. S^J^ , Inasrmch as the expression R-kW 
does not correct adequately for gu^ssingj and has the same 
validity as S ^ ^ for Attempt,^ constant, the loiter be- 
lieves it should be abandoned in favor of the simpler form. 

The following exairrnle illustrates the foregoing dis- 
cussion. 



< 


K 


\N 


i^-W 


K 


/^ 


VA/ 


A-vaA 


1^ 


2 


8 


-6 


-.?c 


9 

**.'-j 


2 


-4 


20 


2 


8 


-6 


^ 







-4 


3^ 


5 


5 


C 





1 


-a 


i 


4^ 


4 


6 


-2 


1^ 











5<^ 


7 


3 


4 


on 


.^ 


-3 


6 


^5S 






::io 










3C 


4 


6 


-2 












^" 


(^~^^^- ^ 


l<-(A->^) 








4^0 


4 


16 


4<^ 


8C 








IOC 


4 


16 


'}■'* 


4^ 











1 


4 


4 « 


ft o 








1 nr 


f*. 















J.-- ■ 






• • 


« • 








400 


9 


36 


er^ 


120 












7^ 


-2^ 


HT^ 








VK-, 


\JlS<^^ 


^ 










I 




-T-t^o 


^_ 











AHa~w)-= ^p^^^;^;^ - »r5 



p. 69 
Section 1;? 31.:rile Is^tlos 
■?.. The Correl'^.tion Bet?;een . Sr>eM «'^'^^- Accm^aoy 

<!■— WWW8—BIII I I H1 III ■!! M M II II J » l u >WW*a— M— >— ilMI W iiB n i l ■ U MlJlll llilia m iilii ■ » i ■ > ' il» IM| IW H lllll ln a 

Tli8 ror-Tolae S^- -% ond "S - ^ "r.;; ')c conveniently 
©riTployoc; to irdex Speod aad Accuracy respsctiTely, The 
latter form has been xised Girtenclirely in the first two 
parts of tliis study and haa been fouBi' .■ave Taluable 

properties Tjfiich other inde:xor; do not ^^oasess. Witii Time 
constimt r.s in the intellirenc© scalGB diacusoed, the for- 
uaia Ijr ^poed roaaoeg to o c c A ; i.e. tiie Atte-iipts 
give the moasure of Speed directly T;7h0n Ti'ia is fi3ced. 
fhen AtterT/tf. ^xq constant, the Speed ie r^iYsn by the re- 
ciproail of the Tinie (in suitable imits). 

Tlio fTtenoral expression for tlie correlation between 
two ratios -^ and ^ may be Ts/ritten in the fonn 



"^ore the Vs are coefficlGnts of variation ^J.YQn by the 
for^rala V - '-^^7^' (Theore-^ " Appendix). For t!i9 two ra« 
tios -^ and -^ this expression becoaes 

For T- const^iiiij Vr - constant, and all correlations vdth 
T are zero, therefore, 

which QXfression reduces to 



p, 70 

Equation (9) thus p;iveg tlie correlation between 
Speed and Accuracy for tests T*iere Time is fixed. The Ila- 
xitnajn Yalue, or AA|-+i-(Joi3 giiren for TIar^+i- . Thus if 
the pupils get every iDrobleiTi that they attempt right, Speed 
and Accuracy mil be r^erfectly co2Tel€i,tGd. For zero corre- 
lations between Atteraptrj and Hic^itSj Speed and Accuracy 
are nG.p^-atiirely correlated, the value ap;":roxlmatins - -5; • 
Finally, if the ratio ^ is equal to th.e value of/lAR , the 
correlation between Speed and Accuracy will be zero. Tables 
5 and 19 indicate that these l?.st relationships will hold 
very closely for intelli/^^ence test data. 'Tlic ratios of the 
Vfi and the corresponding correlations are approximately .7 

For.iTdla (9) is very uaeful for obtaining the correla- 
tion betv/een Speed and Accuracy from the Dingle correlation 
table for Attempts and Eights. This tabic gives the values • 
A^a,^^.^5a, Mr^ca.,^^ (Ma i7hich are all that are required 
to obtain H-aJ . The correlationsiTith Accuracy in Part I 
were co.TTiuted froiii ratios -^ obtained for each variate by 
division. About a third of the coefficients were then check- 
ed by the above foriTiula. Substitution in equation (9) re- 
quires but a few tiioments, liiile the division for ratios a- 
lone takes about an hour for 5^ esses. A great serving of 
time is ^ therefore, effected by the use of the above for- 
mula, especially if the constants in the foriTula axe needed 
for other purposes. 

If Atterupts are fixed, Speed is raeasui-ed by the recip- 



p. 71 

rocal of the Time, and Accijiracy by Rights. Equation (8) then, 
reduces to 

For reasonong test material of the Trae-False type, ad- 
ministered mth Attempts constsnt^ Iott ne,?:r:,tiYe correlations 
were olDtainod for Ri?3;ht3 and Time, indicating that the cor- 
relation between Speed and Accuracy given by the last for- 
mula is positive f-nd low. For 15 .groups of about Z^ impils 
each, the aTerage correlation jIar,- .:i^± ,o5 . For both 

I A 

t^poen of test achiini strati on, then, Speev.1 and Accuracy ex- 
hibit correlation that is zero or ba^rely large enough to be 
sifmif leant. 



\.y 



b. Thg Yalidity of Sin?:le Ratios a?. Scoring: Indexes 

It has Just heen shoim that the ratios giTing Speed 
and Accuracy are relatively independent measures of intel- 
ligence. The choice of the proper index mil therefor de- 
pend upon criteria such as the purpose for ffliich the meas- 
urements isjere mrde, the Taliditv of the index, and its re- 

•■ft' 7 

liability. The question of validity will be taken up first. 
The use of certain linear foruralae has been justified 
on the basis of their validity or correlation witli a cri- 
terion. Tliis same ^-^rinciple may be applied to ratios. If 
a criterion J K, be substituted in Forr.iula (8) in place of 
-^ an exprension for the validity of -^ may be i^^ritten in 
the form 



(1^) Hk^ ^ /lK>cVx-/lKxyY 



\]v^2JLnVxV^,^VY'- 



!Hie correlation tables for H-k^ and. ni<x r.ili furnish all 
of th'^ r'ntn nec9f5sary for this for-uil.'-. ^rK" Bave the labor 
of calcaj-atiiv'^ ii\<^ by the direct iiiet;iou of divisiaii. 
In rpmorsl IfAki is 5ifm.iTics:n.tly hiciiQr thQnJ(lc^ or /Iky 
its ago in place of the si'iroio Tariablss is juBtified. an. 
the !f?proundo of hidi^J^ validity, 

TJliile scoro in not the best measuro of Talidlty for 
indeye-:: ---hich : r^^ ^o olosel^r T'^i-t(^ to it, nevartheless 
the di.ta ia Table 5 'sail givo i su/^r^6.Gtion as to tlis r-sn- 
eral nothod. IHie orcler of tho correlations between R;^; A; 
and HGoi- v^. Ask^.^^, Jlsl^^-'?^ ^ /Ls4^.s<^ * The dif- 
ferf^nces oxe all sir-nlficant horc so tiiat on the basis of 
validity-^ is a better* inde:-: than A, but not so ^ood as 



^■> -, T 



c> The HeliaMlity of 3 i;!rlo ....:. vi;..- .w 3oorinp: Indexes 

Equation ;3) is arsln useful in --irocliGtin,^ the reli- 
ability of p. ro;tio mtliout direct calculation on the ra- 
tios ths:nse?.ires. If ^' pud ^ denote tiie ratios in qucotion 

on succesoiTe trials of -S^lie nar,i© test or b;: pai-allei forros, 
the reliability forrmlc, nk^y be ^Trltton in the forn 

In order to ealciuj^te thin qiientity, foiu" correlation ta- 
bles are rsquireds X/i., X,Y^^ X,X, ^ o— ^ Xi.Ya_ 



p. 73 
If thoy cx^: -^rf-j-naxed all at once, the rn.'^.r.n'i n?>l frequen- 
cies give ezceilent checks on the distributions. As in th© 
case of validity if J^if^u is significantly greater 

than Aa,>^ 0^ /l^,x^^ , it is to be preferred, as 
the more reliable index. This method was applied to the 
Terrnan Scale Forms A and B. The results of the direct cal- 
culation appea.r in Table 7. Predicbion oj Fooiiala (li) 

Accuracy end Rif!;hts are si?«iificently i;aore reliable tlian 
Attempts, but are not essentially different from one another. 

Tlie intelligence quotient, and similar ratios, are es- 
sentially a score divided by a chi'onolo-^ieal age, fihen the 
score is expressed in af^e units and taken from a suitable or- 
igin. The choiee of the age unit is important cliiefly because 
the resulting; ratio is then a pure number end easily inter- 
preted. As fai' as th.e validity aiid reliability of the ratio 
are concemiKij the dioice of the unit for score is of no con- 
sequence , inasmuch as correlation is a measure independent 
of the units eiiiployed for the two variates. The origin from 
Tufeich the scoro is taken is alway" of importance since any 
shift obviously changes the ratio e..<T. ^"^ 't - 
The origin for the score in the intellif^ence quotient is the 
same as for chronological a^e, hence no difficulty is encoun- 
tered. 

Foraila (8) is a function of /Is and Vi , oaly the lat« 



p. 74 
ter beinr affectod. by the orir^in fro'n which the variables 
are taken. Indeed, by suitablo choice of origin all of the 
Vs may be -nade equal, so that the forrrala reduces to 






Lettini<^ ^ and w^ denote a.?^e, end / and Z' scores on suc- 
cessive trials of a test, an expression of the reliability 
of the ratio oi^ nuiy be written in the forn 

n S S f "<;"/ b.s.. -"7 Is, cx^ - yi.s^,c3^-g>g_ 

Furthomioro if j (s.o^ - jU^^e^-p- tiiis expres^3ion re- 
duces to 

Forriula (13) ?;ill have the value +1**^^ for fUs^tMroj and 
will be zero for I -t-KsvSi, ^sJU-^^^^^ , For a civen 

positive value of AsvSx_ , t^e reliability ,^iven by the 
fonmla rail increase as A^-^^-s^ decreases fro^ii the val- 

ue ^ — • 

As an illustration, let As.s^ -+. ^ . Tho funotion 
[1 s, s^ jmy then be t^ibled for the ar^pnent A' 

folloiTSS 






Ih 



1,0 

,i 



vr'^^ 



^ 

•i-,75 

+.83 

+.88 

+.9 

+.91 

+.93 

+.94 

+.94 

+.95 



A*^g-- ,s 



■ t 



+.9 .T 
+.8 

+.6 
+,4 ^ 

+.2 . 

o.n • 

-.2 .:^ 
-.4 ^i 



p. 75 



-.6 

-.8 ■' ' 

-.9 ^ ^^__^_ 

Figure 6 T7hich is based, on the table sIiotts that the high- 
est reliability of the ratio -51^ ocouxs with the hi^- 
est ne'-'-ative Talues of /L^-*^ and decreases as /*-s^>-^ 
increases (to the right). The value for 7Mch j W Ji^ -^^ .^ 
is found by solvinr^ the equation 

giving x^ . 5 5 so that the ratio has a greater reliabil- 
ity than RsiSa, up to the value A^^^ - . 5 , Inas- 
much as .3 is a good reliability coefficient for score, 
and fls-*^.^ is seldom as high as +.5, the suppostitious 
exai'Tple above indicates that ratios mth age have in gen- 
eral a greater reliability than that between crude scores. 
The ne,«;ative correlation usually found betn^een score and 
age for a given grade group (See Tables 8 and 9) also in- 
dicates that ratios of the above tjoQ are most reliable 
?7hen the group is thus selected T^ith respect to age. 

A final exarrrle will be given to illustrate the effect 
of transferring the origin to eliminate the . Assuming 
the apr,roxiraate values \lf\- ^^ -o^, / (ar - . ^ ^\) -o-^ ^ ? 
c3u^ fli-o^^-.v , formula (11) ?/ill give by sirnple substitu- 
tion 



/X,*<fl- /•'«_*'« 



An increase in reliability of .'^S is thus brou^^ht about 

by the transfer of orirdn. For tlie linear form "^-fx R-^cW 
such a Bhift ?all, of coui''n0j h;:iTe no of foot ur^on the re- 
liability or ¥G,lidity of the forrrrala (Sog Theorem 1 Ap- 
pendix) • 

!Ehe bases for selecting a si":ii:;-:le ratio as a mode of 
indexiiip, may be listed as follows 
1. The desirability of indexinr^ another feature of the 
general charactoristiCs e.p;- Accuracys thou<^*)i a sub- char- 
acteristic of iiitellip;©nce is essentially different from 
S-oeed. 

^>o The raiorsl -oropertios of the ratio as compjared with 
other Tarisbles e.g. the intellir^ence quotiorit facilitates 
coniparisons with norrml adiieireiaGnts* 

3. The validity of tlie ratio to bo detesniiined by. Foniiiila 

4. The reliability of tho ratio as deter-iined by formula 
(11). 



Sectio:; 2^ Ceiioral Conclu sions 

1» The various types of response to test material have 
been treated as index variables for the traits in question* 
An analysis of theoe variables for intelli'-enoe tost data 



p. 77 

reTsaled fairly definite relationships bet??een them as 
indicated by the coefficient of correlation. 

S. By eliminating^ the difficulty factor, the -primary 
index Yariables were reduced to AjH,! , and T, one of 
these oQin^ fixed by the method of test adniini strati on. 

3. in analysis of whole scales indicated that all of 

the primary variables have Yalua.ble prooerties as indexes. 
The introduction of the sir.orple ratio ms,de possible a com- 
parison of the indexes revealing; them in order of general 

reliability as SjR,RjW, and Aj mth Time fixed. 

I 

4. According to the criteria employed, the co'T-licated 
formulae used by the authors of the tests, are not signi- 
ficantly better than Rights alone. Acc-uracy comes next as 
a 5;enerally reliable index, and has -oro":erties ?;hich the 
other variables do not possess. For batteries of tests, 
then, scorinr!; by Rights alone is justified by reason of 
greater simplicity and practically equal reliability 
as cojiroared with more com"lieated formulae. 

5. In discriminative capacity, Attem^pts proved to be 
highest and Acciu-acy lowest; i.e. individuals and groups 
differ more widely in Speed than in Accuracy. Lack of 
discrimination between groups is of less consequence in 
en index thsii failure to differentiate bcti^een individu- 
als. Accuracy, therefore, retains its high place as an 



p. 78 

index regardless of the sli/rht inter-group differences 
shoim. 

6. Analysis of the scales by conrponent tests furnished 
a check upon the results obtained for whole batteries. 
The coefficients in p-eneral are lower and less consistent 
than by whole scales. 

7. The validity and reliability of tests are increased 
by pooling coiTtponents, Estiimtions of these correlations 
are furnished by certain predictive formulae , which in 
p;eneral, tend to give over estimation very early in the 
emulated series. 

8. Both theoretical prediction snd actual results in- 
dicate that pooling tests soon ceases to increase valid- 
ity and reliability materially. A battery of five well- 
selected tests is about as satisfactory as one tv.dce as 
long. 

9. The formula S-&-R+c:\>7 i-^^c^ haen slioYin to be the 
most general linear form. Foniiulae for validity have 
been worked out by the method of least squares. 

1^. The linear forTiula above is open to - question because 
of its sensitiveness for values of C. This implies that 
a new deteiraination is necessary for each neis; group 
dealt i^rith. 

11. The gcorin<7 forraila^-v^/ is criticized becauso of its 



p. Id 

failuTQ to correct for r^eosinf^. It has boon shoim that 
for Attempts constant (the asual method ?rl th True-False 
tests) corrective formilae of the tyr-e R-^W have ex- 
actly tli8 saiae validity as Hirjits alone, 

12, Special f orTulae and niothods have boen r/orked out 
for detcrmininr^ the validity and reliubiliby of siraplo 
ratios, 

i3. The valiiahle propertien; possessed by siraple ratios 
indicate that they are hi^dily dosirablo and useful scoriiv^ 
devices in nite of the labor of division. Special tables 
give such quotionts directly. 

14« More conTolicated for-mlae ha,ve not been dealt v?ith be- 

'u 
cause 1J,i0 labor involved in tlieir use would be m-ohibitve 

no '-T^'/'^or iTiiat virtues they r-iir^h'^ '}^' ^:-nd to -obsess* The 

sensitivoness of such forirulae is additional reason for a- 

voiding them. 

15, The results as a whole rioint to the concluaion that 
for batteries, and for sinf?le tents as well, tiie snost de- 
sirable and useful indexes are K and B, 

I 



Bibliography 

Memoirs of the National Acadeiny of Sciences, Yol. X?, 
Psycholop;ical Testing in the United States Array. 

CSpoannan, Correlation of Sum-'? and Differences, British 
Journal of Psyoholoi^, Vol. ?, vv* 41-^-^1^6. 

A.S.Otioj An Absolute Point Scale for the Group Measure- 
ment of Intelligence. Journal of Educational Psychology'', 
Yol. IX, Nos.S-and 6, Hay-Jane, 1918 

L.L.ThuTGtone, A Scoring Uethod for Mental Tests, Psy. 
Bull., Yol.XYI, No. 7, July i9i9. ^ 

T.L.Kelley, The Reliability of Test Scorer. Jl.Ed.Research 
Yol. Ill, IIo.6,p.379,May 1921 

A.S.Otis GJid H.E.l^ollin, The Reliability of the Binet 

Scale HY^ p6da{50gioal Scalec, Jl.Sd. Hptsoarch., Yol. IY» No. 2 

121, Sept. I9n 



• Biblior^&r^y — Terts 

^MJ.'oil • ■ , ■'• " .r ^ ;tatisticB,. 

Chaxlss Griffin and Comnany, London, 'Ui-J* 

Sons, Hew York, 1920 

D.G.JoneSj A First Course in Steti sties, a.Bell imd Sons 
London. 1921 

W.Bro"^ piv! C.H.ThoriTOBon, 'Hie Ks^^entials of Mental 
Aleas'xro.teuu, GaTibrid^^e univeraioy Pre;^;^, London, i92i 



APPENDII A 



Correlation Tables for Reliability Coefficients 



COHRELATIOI'I TA3LS FOR SCORE ON TEB2IAII FC: 
TEE!.li^J FOEl 3 VflTH aEOUP I HIC!!! C 



"./ITil SCORS ON 





SCO 




Oil TEmiM 


FOK^ 


B ( 


FIRST TRIAL) 


















?j 


<5- 

co 


&- 

::}- 




1 




<5~ 




0-. 
o 

1 


C^ 


fvl CO 

1 T 


<>- 


3-^ 


0^ 




) 


5 

H 




1 


; 



en 


1 
o 



In 


o 


o 


1 

o 

6o 


o 


1 


o 


5 £2 




o 


O 


I-' tS 


a 


^or..2?^9 . 




• • • 




O • 


« e « 


« » « 


9 » 9 


9*9 


9 


» 9 « 


1 

> 9 9 tt e> 9 


ft « 9 


• e • 


« • • 


...1. 


a.l 


<i 


19--192 . 




o « » 




• » • 


• • • 


o a 


9 9 6 


9 9 « 


• 99 


1 9 e 


) 9 o a O 9 


ft 9 9 


I 


a a i> . 


aa. 


1 ) 


lo -IGv:' • 








• to 


« 9 « 


a • • 


9 9 9 


9 ft 9 


» 9 ft 


1 O 9 


ft 9 ft 


» b • » O 


> • • {a s 


• a u • 


a a • 


'ij 


170-17- . 








« • « 


• • « 


9 9 


9 O » 


• f> 


• 49 


t 9 « 


) 9 9 


1 • • 


• 9 9 


I 


. a ; a - 


• a • 


(/J 


16^-16;? . 








« a « 


« • « 


• » o 


O 9 • 


9 9 


n 9 Q 


1 9 9 


ft B e 


b • • 


t • « 


.i. 


kl* 


..^.. 


..1 


<r 


IS'^-ISS . 








6 <t * 


• C » 


6 O C 


» • » 


• » « 


* 4 


» » 


> « « 


.1. 


,2. 


o 

» >'U 9 


.1. 


. . !> a ' 


a. 6 




14^-149 . 








• « • 


« 6 « 


ft • « 


9ft* 


9 9 9 


.2. 


3. 


' ft 9 


> » . » 'J ♦ 


a. 




» a • a ( 


..8 


S. 


13^-130 . 








O <» 


• 9 « 


O • ft 


9 9 9 


.1. 


kl. 


.1. 


.3. 


.2. 


1 « • 


I , . 






0,8 




l?^-!-^.^ , 








.1. 


• <* « 


9a 


1 


.3. 




1 .i 9 


» 9 e 


.1, 


1 9 9 






> • i> • 1 


.10 


tL 


li^-l.U . 








• Ck A 


.1. 




ft-'^. 


.5. 




»'-i9 




» • • 


1 a • 


1 W 






> • i> a < 


.16 


-7 


i^^'-i^i? , 








« O 4 


• « • 




.7. 


.7. 


-? 


k e f< 


• o © 


i c » 


■ • 9 








.17 


<r 


yO- 99 . 






.1. 


BOA 






.5ft 


1 
ft «k « 


► a » 


» A 1. 


> « r. 


, « o 


1 a b 






> t> a a « 


.13 


S 


80- 8Q . 






^1 


•-> 


.6. 





9,^9 


9 9 9 


» .1 a 


I- * » 


i> -•. 


} 4 4 


1 i» a 








.16 


oi 


70- ?•;} . 




.4. 






n 


.3. 


.1. 


9 a 


» « 9 


1 a ft 


> o « 


. r. o 


r c e 






> « a e f 


.16 




60- e'i? . 




.3. 


.1- 




.1. 


.le 


e • a 


« • 


i 1 


B a 4 


' e 


1 a A 


1 O a 






r o a a < 


..8 


50- 5';? . 







.2. 


« ft A 


o • « 


• 99 


• « a 


9 9*' 


* V 


» 9 9 


1 ff 9 


V <3 e 


> 9 


> . . i> o a 


..ia. 


..4 


2 


40- 49 . 


.1. 

• X. 

,1. 


.1. 


o 


























..4 


O 










- • • 














1 


• 9 i • 1 


,.1 




n « ft 


6 o « 


• • • 

« o e 


9 4 


« * 9 


• • • 

. . . 


• 9 « 


ft 9 « 


t * V 


» « o 


. » 9 


1 • '1 




> a f a ' 


..1 


o 

1/0 


f 


3 


10 


10 


ir^- 


13 


14 


30 


VI 


9 


9 


3 


4 


4 


4 3 


- 1 


135 



PL- -h. ^o-gi. 



O [ o 






Ma-A^B^ n,43i2..4,5" 






grdn is significant 



diff.may be significant 



(r 






TABLE FOR A5?Tn',!PTS Oil TEKlAIs FOHLI A WITH 
ON TEK-IAN FOR!] B ITITII GRO.'? I Iliail C 







-1. \J^u. 


a B 


:first t. 


■T '■r \ 






< 




<ir- 


<s- 


'>- 


o- 


a- 


o- 


o- 


o- 


Or- 


o- 


C3- 






c>3 

1 


i 


o 

T 


l 


1 


S2 

1 


( 


1 




1 


1 


f 


u. 




s 




g 


o 


5 


o 

CO 




O 


o 




o 


§ 




< 


,» - ,\ -i -> 
















«» 


,. 


n 


«1 


2<> 






1 » • 1 
» * • ' 


« » 1 


I • • 


o « • 


• • • • 


1 u II « 








on 


16f^«iCv? 


> o a 1 


« « 


' A a 


1 


..4 


. .;J. 


..4. 




L«6. 


► .4. ,..,.. 


• a 31 


l- 


15^-1').' 


.U, 


• o 


1 « • 


1 • vy 




» • ■> » 


>«•.-• 


• • %' « 


1 

to ft/* » 


» « ft « 


' 9 « O O 




2 


14^.^;-/ 


» O » 1 


tt p 


1 


• • >. > 


• • • < 


1 A ^« • 


• • «i « 


1 • '' » 


» <1> o o 


^ « tr * 




9 • ■Ui^ 


o 


a.3^-.i... 


h • ci> ' 


« « 


» * <> 


.A 


> • :^ • • « • 


> • V » 


>.x* 


h O A o 


» «> n tt 


,.,ft.,l.,.D 




ir-i:. 


► • o 


• ♦ 


la 


..i 


L.i 


1 a O « 


• ftiiv 


^ o o « 


L... 


» • o ^ 




...4 


t- 


un-i.ij 


» « « ' 




,u 


,.1 


..i 


1 • > ' • 


I 9 * « 


» « « » 


» • » ^ 


► *•**» 


..» .| «««0 


e. 


ic^-i-.^ 


k s c < 


« • 


» o e 


» « 


a a ft 


1 • 9 


> » « o 


..1, 


» C O 17 


? w • » 


• • * « 1 


• • •!> 


^ 




* « • 


.1 


1 • • 


' « • 


• • • 


1 « * • 


• « « O 


f a «. o 


* O O » 


? # (» » 


> • « W 


« & « i& 


H 
1- 


8^» .Jy 




e o 


t * u 


' m • 


a 9 a 


1 « • 


W • * 9 




! 9 a o 


E> <> • tr 


ft » fc to 


s « » » 


1 • 9 ail 


< 


f 


3 




4 


l^ 


li 


18 17 


sx 


gr 


Ivt- 


» 


135 



jl^ +,t -^^ i .032_ 









CSi~^^^ 3.<^q i '-^^ 



r • 



^in it: sifpificsnt 



/.diff. riiay Ij& sigaificaat 



COimSLATIOlI TABLE FOR KIQIFTS ON Tmm 
ON TMIAN FOR-' B ^-'IT^' ffflO^F 



o 

Z. 



o 

Co 
f- 

5 



HIGil C 



ixiui..j.i* tj.; i'^jiuiAi, 



-i ii vi'liUji' li' 



ilAL) 





1 


1 


1 


<3- 



1 


1 




\ 


>- 
O 

1 


(3- 

T 




C3^ 


f 


T 


'J * 
1 


r 




o 


% 





o 




^ 


o 


§ 


o 


^ 


CO 


2 


'2 


o 




17^-172 


• • • 


ft « » 


« o 


U 4 3 


• « « 1 


« * • 


a • f* 


w » o 


» » i 


O 9 » 


* » « 


» ft ft 


» ft o d 


ft! 


ft • it • 


16^'-169 


• ft » 


• « a 


4 


<ft <* 


4 « « 


• • « 


^ • a 


* » « 


& • O 




« a « 




ft ft < 




1 » ft«»« 


IS'^-l'v.j 


% 4 9 


• t> • 


» • * 


• • < 


»<?>•« 


■» t '3 


» • » 


» « « 


... 


ft *» ft 


a a « 


■ ft A 


■* "i » 




..a 


i4^-:.-w 


• 00 


a 9 w 


« » « 


• a * 


» O « 4 


e « o 


» • » 


» • A 


n> • a 


ft ft » 


• « v^ 


» ft '■■* 


ft ft • 




..§ 


l^^-lo^ 


• * • 


• • s 


« i> a 


• » » 


9 rt o -. 


» » 


» O ft 


.a 


..3 




•Ct 


1 • a^ 


ft ft I 




,ac 


i.r-.i':p 


A « 


» tt n 


o a A 


• « e 


«X « 


« ft » 




..6 


..;«i 


ft ft 


ft.X 


> • ft 


' ft c ft 




.14 


ii^-iiQ 


» o 


fr « « 


V a « 


• '* 


» 4 *^ fl 




t • '-. 


.•4.0.4 


• a ft 


• o • 


» ft ft 


ft «■ « 




..15 


ir'O-i^^tj 


• • • 


• • • 


• « 


s • « 


• ..a 




a: 


..4 


l.X 


« • • 


ft » • 


> • « 


« ft • 




• 26 


y^- dd 


O O r» 


» » » 


p t. -^ 


> • •• > 


6 * ' »* 




■■'"* 


ft M A 


><•»(>•• 


» • •? 


» ft 1 


ft « ft 




,.r/ 


3^' Q9 


ft O o 


..r^ 


• •u 


» •=.-» 


K •'^V« 


• '-a- » 


» « 


» O « 


> o ft « » • 


« • » 


> (1 ft 


• ft o ft 




.a^ 


r. 79 


.u 


,a 


• •6 




1 • * « 


« a » 


k « » 


• 9 •> 




• « 


> • • 1 • ft a 




>a4 


6^« G9 


.•1 


...-s 


..3 


1 M 9 


|t • • 4 


» » « 


» • o 


»«9I0««<»«« 


m » '.■» 


S A ft 


a a a 




.,.6 


13^- ;)^ 


^ 4 •> 




1^ 


» • .9 


» r» '> 4 


■> « © 


>'••<' 


1 <>«^a«t»»* 


I* ^« 


> • « 


1 » .» ft 




ft, 3 


^- 4^ 


» ■ « 


» • • 


» • • 


» » • 


B • « Q 


• e « 


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1 
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.a 


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k O <3 4 


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► ■» * 


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> ft • 


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r 9 * 


• •« ' 


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..1 


f 


L_ 


-f^ 
Cj 


16 


-t '1 


X3 


ll»>f^ 




m 


12 




/ 


,d 


eft 


Z 


135 



/1_^ -H, S^ t ii , O I \ 



(Ma ^ (oz.4-( dt (. i7 



GX = -^3. 5^-± 0,17 
.^^-<5;:= 3.4-1 =L /.M-7 



/, 5^J.n is slnnificsnt 



,', difft my be si^ificant 



d 



coBEEyiTioii TABLE FOR micms ON tseia:i foe.1 a with 

^OHSS m FOM B WITII OSOIJP I IHG^I C 







t3lOI^ 


3 0' 


] TEEMAH FOE' B (FIRST 


TRIAL) 






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.:3^ 


"T~ 


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tj~ 


a— 


a— 


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% 






1 


1 


1 


1 


1 


1 


f-^ 


■>3 

1 




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o 

-7 






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tt o « 


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1 « « «<* « 


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90- 


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sr- 


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ft 9 a 


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9 /r£ » f» 


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52 


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2. 
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19 


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f 




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. 139 



7L- -h. 13-7 ±,,0 3.7 









,\ loss is ins i{?!niif leant 



/, diff. i-^ in«i??iific^t 



Ji- 



COBRELATION TABLE FOR ACCURACY Oil TEK5AN FORLI A WITH 



ACCUHACY 



GiiOIiP I HIGH C 



AccuEACY m 'iimm Fom b (fibst trial) 



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85-39 




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1- 


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60.64 




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55-55 




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45-4i^ 




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19 9* 




9. .12 


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..I 


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1 t 


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k 9 9 9 6 9 


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35-39 




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1 


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► • • 


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9.9.1 


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13 


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■J 

i.v.> 


A ■ 


<-, 


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135 



/L - f.'^ifi i- -on 



[V\/v-_ 0. b3.3 ±0, (>07 



r^A~i^a- o,of^± 0,0 II 






,', gain is cigjiificent 



, ; dif f . its not significant 



APPENDn B 

"■Dieorens Relnjdn£f to Oorrelr.tion 

Notations 
Xy ^ ^ 2 - - irririablQs fron arbitraiv oririns 

>; ^^^ variables from respectiYG metyas 

l^>, risfinB of t?io variables X, 

G^ st:ndaxd deTiationn of X^ . 

V,.^ PearBon'n coofficient of Tra-iaMlitrr, -^^^ 

JU^ - - - - product monont correlation 

Z - sum of Buch (iuantiti©3 as . . 

N frequGncjr of the popiiJ.ati on 

o~,h-,c, — constf 



?heoro:!i. I* Tho correlr.tion bct-r/asn trio 7?.ri able3 is the 
sa^. a3 that bntrcfm any two linear faietionc of eg.ch of 
them i.G. 

Tra-isferrini-^ the variables to their respective momiBf 

/l(aX + jt)^c^ -hdl) = jTl^jU"^) 




TlieorOiTi 2, The correlatio n bo two en tiro ratios -^ eM. — - 
" — "^ — ^ — *~~~ — ^ — — -^ — ^ — ~~- — ^ -~-~ saT 

is ^iveii by th e for.mila.. 



^ w 



Ihe nieRns mid ntandarcl deviations of -^ are given by 



b. 



a 
equations (9) and (i^^) in Yule, 



The renuired correlation J Ix^ mil then be given by 



Expsuidinp;, ne^clecting terms higher then the second degree 
and substituting forai (a) and (b) gires (2). 

Formula (2) gi^es satisfactory results for fairly 
long series, but for very short ones, considerable error 
occurs due no doubt to ner:l0cting tlie higher po?/ers in the 
expansions of the binomals. 

Theorem 3, If X - tj^"^ + -b- Z. , the -partial correlations 
between the variables may be ?Titten » 

J (.y:^- > - - I. 

as is evident by inspection. 

IbcaiTiple: Since A -i^t^ » J ^(^w --' s if A-G&m/i^, 

a. S.li.Mo, xntroctiction tv tHo"'TC;'ary'"or;Jtati sties, 



Theore'n 4<, Tho correlation betijoon ?. variable and. the Siom 

a 
of n othe rs is p;iven by the forniiia , 

(this orpression is a special ca.se o.f .jioarrrrji's General 
ForTdla) 



Since ^^ xi+ ^^ -r - - - ^) — \p?^- ^.^^ - ^,^-«-:L(nx,^,t>x, b-^^ ^ J^>:fi7^ 
and ^x-j - A//l/tY'^^ J tlie ri.n;hu hand ;neiaber above 
roduces to the ejr^srosaion in (4) 

Corollar;/. If -t^i? staidard deviationo Oo^ arc all Qgual 

Theorem 5« The /3Qi^rela.tion bet?^eeii tlie 3it:i..ofn variables 
aiid n otiior vari ubiss is p:ivon...by.,.-bi:ie fQrr;Mla 

Wae proof is similar to (4) 

Coroll?^y X, If tho str';t¥'iaKl devia,tioii3 g^ ixe all equal 

/•>.* n _ / u,x,' H-/(-A.xj. -t - - - ^T*^^ xia^^a^ 



--h-n. 



where A>i>v ■lonot--^ th^ loft h-vi^l ■■lo-ibor of (5l « 



a. G.3i^'eaiii^a, British Joixmal of Psycholofry. X9i3p Yol.?, 

b. G»Spearm?aas, loc. ^t» 



Corollary 2. If the oorrelationg flry. ire all equal , 
equation (6) may be TTritteii 

(7) /I... 7^, 

This iz Brovii's Theorem, but an sho'sm aboTe it is merely 
a special case of Speaiinaii's General Formula. 



* :'<^ 




